10 9

10_8

10_10

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 9's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 9 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X12,3,13,4 X2,15,3,16 X14,5,15,6 X4,13,5,14 X18,10,19,9 X20,12,1,11 X8,18,9,17 X10,20,11,19
Gauss code	1, -4, 3, -6, 5, -1, 2, -9, 7, -10, 8, -3, 6, -5, 4, -2, 9, -7, 10, -8
Dowker-Thistlethwaite code	6 12 14 16 18 20 4 2 8 10
Conway Notation	[5113]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 3, Braid index is 3		[{2, 12}, {1, 7}, {11, 6}, {12, 8}, {7, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 9}, {8, 10}, {9, 11}, {10, 1}]

[edit Notes on presentations of 10 9]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["10 9"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X6271 X16,8,17,7 X12,3,13,4 X2,15,3,16 X14,5,15,6 X4,13,5,14 X18,10,19,9 X20,12,1,11 X8,18,9,17 X10,20,11,19

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -6, 5, -1, 2, -9, 7, -10, 8, -3, 6, -5, 4, -2, 9, -7, 10, -8

In[6]:=

DTCode[K]

Out[6]=

6 12 14 16 18 20 4 2 8 10

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[5113]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(3,\{1,1,1,1,1,-2,1,-2,-2,-2\})}

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 3, 10, 3 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{2, 12}, {1, 7}, {11, 6}, {12, 8}, {7, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 9}, {8, 10}, {9, 11}, {10, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 8.2941 A-Polynomial See Data:10 9/A-polynomial

[edit Notes for 10 9's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} Rasmussen s-Invariant -2

[edit Notes for 10 9's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+3t^{3}-5t^{2}+7t-7+7t^{-1}-5t^{-2}+3t^{-3}-t^{-4}$
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8-5 z^6-7 z^4-2 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 39, 2 }
Jones polynomial	$q^{7}-2q^{6}+3q^{5}-5q^{4}+6q^{3}-6q^{2}+6q-4+3q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-7z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-17z^{4}a^{-2}+5z^{4}a^{-4}+5z^{4}-16z^{2}a^{-2}+7z^{2}a^{-4}+7z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+2z^{8}a^{-4}+2z^{8}+2az^{7}-2z^{7}a^{-1}-2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-18z^{6}a^{-2}-7z^{6}a^{-4}+2z^{6}a^{-6}-8z^{6}-8az^{5}-2z^{5}a^{-1}-4z^{5}a^{-5}+2z^{5}a^{-7}-4a^{2}z^{4}+31z^{4}a^{-2}+13z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+10z^{4}+7az^{3}+4z^{3}a^{-1}+5z^{3}a^{-3}+4z^{3}a^{-5}-4z^{3}a^{-7}+3a^{2}z^{2}-22z^{2}a^{-2}-8z^{2}a^{-4}+z^{2}a^{-6}-2z^{2}a^{-8}-8z^{2}-az-2za^{-1}-2za^{-3}+za^{-7}+4a^{-2}+2a^{-4}+3$
The A2 invariant	$q^{8}+q^{4}+q^{-2}-q^{-4}+2q^{-6}-q^{-8}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
The G2 invariant	$q^{46}-q^{44}+2q^{42}-3q^{40}+2q^{38}-2q^{36}-q^{34}+6q^{32}-8q^{30}+10q^{28}-9q^{26}+5q^{24}+2q^{22}-10q^{20}+16q^{18}-16q^{16}+14q^{14}-4q^{12}-5q^{10}+15q^{8}-14q^{6}+12q^{4}-4q^{2}-4+10q^{-2}-9q^{-4}+3q^{-6}+6q^{-8}-10q^{-10}+14q^{-12}-9q^{-14}-2q^{-16}+8q^{-18}-16q^{-20}+18q^{-22}-17q^{-24}+7q^{-26}+3q^{-28}-13q^{-30}+18q^{-32}-18q^{-34}+11q^{-36}-3q^{-38}-6q^{-40}+10q^{-42}-12q^{-44}+8q^{-46}-5q^{-50}+6q^{-52}-4q^{-54}-2q^{-56}+7q^{-58}-8q^{-60}+7q^{-62}-4q^{-64}-q^{-66}+6q^{-68}-8q^{-70}+11q^{-72}-8q^{-74}+6q^{-76}-q^{-78}-2q^{-80}+5q^{-82}-8q^{-84}+10q^{-86}-7q^{-88}+4q^{-90}-4q^{-94}+6q^{-96}-6q^{-98}+5q^{-100}-3q^{-102}+q^{-106}-3q^{-108}+2q^{-110}-q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 10_9.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+q^{3}-q+2q^{-1}+q^{-7}-2q^{-9}+q^{-11}-q^{-13}+q^{-15}$
2	$q^{22}-q^{20}-q^{18}+3q^{16}-q^{14}-3q^{12}+4q^{10}+q^{8}-5q^{6}+2q^{4}+3q^{2}-4+q^{-2}+4q^{-4}-2q^{-6}-q^{-8}+3q^{-10}+q^{-12}-2q^{-14}-q^{-16}+4q^{-18}-3q^{-20}-3q^{-22}+5q^{-24}-2q^{-26}-2q^{-28}+3q^{-30}-q^{-32}-q^{-34}+2q^{-36}-q^{-38}-q^{-40}+q^{-42}$
3	$q^{45}-q^{43}-q^{41}+q^{39}+2q^{37}-q^{35}-4q^{33}+q^{31}+6q^{29}-7q^{25}-3q^{23}+8q^{21}+7q^{19}-6q^{17}-9q^{15}+2q^{13}+9q^{11}+2q^{9}-9q^{7}-6q^{5}+5q^{3}+10q-q^{-1}-9q^{-3}-q^{-5}+11q^{-7}+4q^{-9}-9q^{-11}-3q^{-13}+8q^{-15}+5q^{-17}-6q^{-19}-5q^{-21}+2q^{-23}+5q^{-25}-6q^{-29}-6q^{-31}+7q^{-33}+6q^{-35}-4q^{-37}-9q^{-39}+2q^{-41}+9q^{-43}+q^{-45}-4q^{-47}-3q^{-49}+2q^{-51}+3q^{-53}+2q^{-55}-3q^{-57}-3q^{-59}+3q^{-63}+q^{-65}-2q^{-67}-2q^{-69}+q^{-71}+2q^{-73}-q^{-77}-q^{-79}+q^{-81}$
4	$q^{76}-q^{74}-q^{72}+q^{70}+2q^{66}-3q^{64}-2q^{62}+3q^{60}+q^{58}+6q^{56}-6q^{54}-9q^{52}+2q^{50}+5q^{48}+16q^{46}-3q^{44}-17q^{42}-10q^{40}-3q^{38}+25q^{36}+14q^{34}-8q^{32}-17q^{30}-21q^{28}+13q^{26}+19q^{24}+13q^{22}+q^{20}-24q^{18}-11q^{16}-5q^{14}+14q^{12}+27q^{10}-16q^{6}-32q^{4}-6q^{2}+36+28q^{-2}-39q^{-6}-27q^{-8}+25q^{-10}+36q^{-12}+11q^{-14}-30q^{-16}-29q^{-18}+11q^{-20}+26q^{-22}+11q^{-24}-17q^{-26}-20q^{-28}+2q^{-30}+18q^{-32}+9q^{-34}-7q^{-36}-15q^{-38}-10q^{-40}+9q^{-42}+16q^{-44}+14q^{-46}-11q^{-48}-32q^{-50}-7q^{-52}+20q^{-54}+41q^{-56}+9q^{-58}-44q^{-60}-34q^{-62}-q^{-64}+50q^{-66}+36q^{-68}-25q^{-70}-39q^{-72}-28q^{-74}+27q^{-76}+43q^{-78}+4q^{-80}-20q^{-82}-35q^{-84}+26q^{-88}+15q^{-90}+2q^{-92}-24q^{-94}-10q^{-96}+9q^{-98}+11q^{-100}+10q^{-102}-11q^{-104}-8q^{-106}-q^{-108}+3q^{-110}+9q^{-112}-3q^{-114}-3q^{-116}-2q^{-118}-q^{-120}+4q^{-122}-q^{-128}-q^{-130}+q^{-132}$
5	$q^{115}-q^{113}-q^{111}+q^{109}-q^{101}-q^{99}+3q^{97}+4q^{95}-q^{93}-4q^{91}-6q^{89}-4q^{87}+4q^{85}+14q^{83}+11q^{81}-4q^{79}-17q^{77}-22q^{75}-8q^{73}+18q^{71}+35q^{69}+24q^{67}-9q^{65}-37q^{63}-42q^{61}-14q^{59}+29q^{57}+54q^{55}+37q^{53}-9q^{51}-46q^{49}-52q^{47}-21q^{45}+24q^{43}+48q^{41}+38q^{39}+10q^{37}-20q^{35}-37q^{33}-36q^{31}-22q^{29}+12q^{27}+47q^{25}+60q^{23}+35q^{21}-28q^{19}-87q^{17}-87q^{15}-9q^{13}+87q^{11}+122q^{9}+61q^{7}-63q^{5}-146q^{3}-107q+29q^{-1}+142q^{-3}+140q^{-5}+18q^{-7}-124q^{-9}-155q^{-11}-47q^{-13}+97q^{-15}+148q^{-17}+71q^{-19}-63q^{-21}-133q^{-23}-75q^{-25}+40q^{-27}+105q^{-29}+69q^{-31}-20q^{-33}-78q^{-35}-59q^{-37}+12q^{-39}+58q^{-41}+40q^{-43}-8q^{-45}-42q^{-47}-35q^{-49}+5q^{-51}+34q^{-53}+38q^{-55}+8q^{-57}-33q^{-59}-52q^{-61}-34q^{-63}+18q^{-65}+74q^{-67}+79q^{-69}+5q^{-71}-91q^{-73}-119q^{-75}-52q^{-77}+86q^{-79}+170q^{-81}+104q^{-83}-65q^{-85}-189q^{-87}-156q^{-89}+16q^{-91}+184q^{-93}+191q^{-95}+34q^{-97}-150q^{-99}-194q^{-101}-73q^{-103}+95q^{-105}+175q^{-107}+99q^{-109}-50q^{-111}-134q^{-113}-99q^{-115}+9q^{-117}+91q^{-119}+85q^{-121}+11q^{-123}-55q^{-125}-65q^{-127}-21q^{-129}+32q^{-131}+47q^{-133}+21q^{-135}-14q^{-137}-33q^{-139}-23q^{-141}+8q^{-143}+24q^{-145}+18q^{-147}-16q^{-151}-17q^{-153}-5q^{-155}+11q^{-157}+14q^{-159}+5q^{-161}-4q^{-163}-9q^{-165}-7q^{-167}+q^{-169}+7q^{-171}+4q^{-173}-2q^{-177}-4q^{-179}-q^{-181}+2q^{-183}+2q^{-185}-q^{-191}-q^{-193}+q^{-195}$
6	$q^{162}-q^{160}-q^{158}+q^{156}-2q^{150}+2q^{148}-q^{144}+5q^{142}+2q^{140}-2q^{138}-9q^{136}-2q^{134}-3q^{132}+15q^{128}+14q^{126}+6q^{124}-17q^{122}-15q^{120}-22q^{118}-16q^{116}+21q^{114}+39q^{112}+42q^{110}+4q^{108}-17q^{106}-55q^{104}-69q^{102}-22q^{100}+34q^{98}+85q^{96}+70q^{94}+49q^{92}-31q^{90}-104q^{88}-107q^{86}-58q^{84}+40q^{82}+89q^{80}+134q^{78}+78q^{76}-22q^{74}-99q^{72}-125q^{70}-71q^{68}-23q^{66}+74q^{64}+99q^{62}+79q^{60}+32q^{58}-9q^{56}-25q^{54}-81q^{52}-72q^{50}-76q^{48}-38q^{46}+33q^{44}+131q^{42}+197q^{40}+117q^{38}-7q^{36}-189q^{34}-289q^{32}-238q^{30}-7q^{28}+269q^{26}+379q^{24}+313q^{22}+22q^{20}-311q^{18}-497q^{16}-368q^{14}+q^{12}+358q^{10}+553q^{8}+407q^{6}-5q^{4}-445q^{2}-598-379q^{-2}+52q^{-4}+484q^{-6}+620q^{-8}+353q^{-10}-157q^{-12}-534q^{-14}-558q^{-16}-252q^{-18}+225q^{-20}+554q^{-22}+495q^{-24}+98q^{-26}-314q^{-28}-485q^{-30}-349q^{-32}+11q^{-34}+350q^{-36}+410q^{-38}+176q^{-40}-125q^{-42}-302q^{-44}-267q^{-46}-62q^{-48}+167q^{-50}+236q^{-52}+121q^{-54}-40q^{-56}-142q^{-58}-130q^{-60}-36q^{-62}+77q^{-64}+113q^{-66}+53q^{-68}-34q^{-70}-93q^{-72}-81q^{-74}-20q^{-76}+71q^{-78}+127q^{-80}+106q^{-82}+5q^{-84}-124q^{-86}-191q^{-88}-163q^{-90}+207q^{-94}+327q^{-96}+231q^{-98}-48q^{-100}-325q^{-102}-461q^{-104}-290q^{-106}+115q^{-108}+502q^{-110}+577q^{-112}+264q^{-114}-220q^{-116}-629q^{-118}-639q^{-120}-216q^{-122}+367q^{-124}+697q^{-126}+573q^{-128}+125q^{-130}-429q^{-132}-678q^{-134}-479q^{-136}+8q^{-138}+431q^{-140}+535q^{-142}+349q^{-144}-63q^{-146}-371q^{-148}-399q^{-150}-187q^{-152}+84q^{-154}+234q^{-156}+262q^{-158}+102q^{-160}-70q^{-162}-146q^{-164}-118q^{-166}-42q^{-168}+12q^{-170}+77q^{-172}+48q^{-174}+11q^{-176}-3q^{-178}-2q^{-180}-4q^{-182}-24q^{-184}+2q^{-186}-17q^{-188}-17q^{-190}+q^{-192}+22q^{-194}+24q^{-196}+4q^{-198}+15q^{-200}-16q^{-202}-23q^{-204}-19q^{-206}-q^{-208}+9q^{-210}+7q^{-212}+26q^{-214}+5q^{-216}-4q^{-218}-14q^{-220}-10q^{-222}-8q^{-224}-4q^{-226}+15q^{-228}+7q^{-230}+7q^{-232}-3q^{-234}-3q^{-236}-7q^{-238}-7q^{-240}+5q^{-242}+2q^{-244}+5q^{-246}-3q^{-252}-3q^{-254}+2q^{-256}+2q^{-260}-q^{-266}-q^{-268}+q^{-270}$

A2 Invariants.

Weight	Invariant
1,0	$q^{8}+q^{4}+q^{-2}-q^{-4}+2q^{-6}-q^{-8}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
1,1	$q^{28}-2q^{26}+4q^{24}-8q^{22}+15q^{20}-20q^{18}+30q^{16}-38q^{14}+45q^{12}-50q^{10}+50q^{8}-46q^{6}+31q^{4}-14q^{2}-2+30q^{-2}-50q^{-4}+68q^{-6}-76q^{-8}+82q^{-10}-84q^{-12}+74q^{-14}-62q^{-16}+46q^{-18}-32q^{-20}+18q^{-22}-2q^{-24}-4q^{-26}+17q^{-28}-18q^{-30}+18q^{-32}-22q^{-34}+22q^{-36}-26q^{-38}+20q^{-40}-22q^{-42}+25q^{-44}-20q^{-46}+18q^{-48}-14q^{-50}+11q^{-52}-8q^{-54}+4q^{-56}-2q^{-58}+q^{-60}$
2,0	$q^{24}+q^{18}+q^{16}-q^{14}-q^{12}+2q^{10}+q^{8}-2q^{6}+2q^{2}-1-3q^{-2}+q^{-4}-q^{-8}+q^{-10}+3q^{-12}+q^{-14}+q^{-16}+4q^{-18}+q^{-20}-3q^{-22}-q^{-24}+q^{-26}-2q^{-28}-q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-q^{18}+q^{14}-2q^{12}+q^{10}+2q^{8}-q^{6}+4q^{4}+3q^{2}-2+3q^{-2}+2q^{-4}-5q^{-6}+q^{-10}-3q^{-12}-2q^{-14}-2q^{-20}+5q^{-24}-q^{-26}-q^{-28}+6q^{-30}-q^{-32}-3q^{-34}+3q^{-36}-3q^{-40}+q^{-42}-q^{-46}+q^{-48}$
1,0,0	$q^{9}+2q^{5}+2q-q^{-1}+q^{-3}-q^{-5}-q^{-11}-2q^{-15}+q^{-17}-q^{-19}+2q^{-21}+q^{-25}$
1,0,1	$q^{34}-2q^{32}+3q^{30}-3q^{28}+q^{26}+4q^{24}-7q^{22}+13q^{20}-6q^{18}+2q^{16}+9q^{14}-17q^{12}+16q^{10}-14q^{8}+5q^{6}+4q^{4}-11q^{2}+22-17q^{-2}+19q^{-4}-13q^{-6}+3q^{-8}-2q^{-10}-7q^{-12}+8q^{-14}-12q^{-16}+16q^{-18}-18q^{-20}+28q^{-22}-22q^{-24}+16q^{-26}+2q^{-28}-17q^{-30}+26q^{-32}-28q^{-34}+22q^{-36}-6q^{-38}-4q^{-40}+10q^{-42}-6q^{-44}-5q^{-46}+6q^{-48}-7q^{-50}-7q^{-52}+12q^{-54}-12q^{-56}+12q^{-58}-2q^{-60}-4q^{-62}+9q^{-64}-9q^{-66}+6q^{-68}-q^{-70}-3q^{-72}+3q^{-74}-2q^{-76}+q^{-78}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{22}+q^{16}-q^{12}+q^{8}+q^{6}+4q^{4}+3q^{2}+3+4q^{-2}+3q^{-4}-q^{-6}-4q^{-8}-3q^{-12}-6q^{-14}-4q^{-16}+q^{-18}-5q^{-20}-2q^{-22}+3q^{-24}+5q^{-30}+5q^{-32}+3q^{-36}+4q^{-38}-4q^{-42}-3q^{-48}-q^{-50}+q^{-52}+q^{-58}$
1,0,0,0	$q^{10}+2q^{6}+q^{4}+2q^{2}+1+q^{-4}-2q^{-6}-2q^{-10}-2q^{-14}-q^{-18}+q^{-22}+2q^{-26}+q^{-30}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-q^{18}+2q^{16}-3q^{14}+4q^{12}-5q^{10}+6q^{8}-5q^{6}+6q^{4}-3q^{2}+2+q^{-2}-2q^{-4}+5q^{-6}-8q^{-8}+9q^{-10}-11q^{-12}+10q^{-14}-10q^{-16}+8q^{-18}-6q^{-20}+4q^{-22}-q^{-24}-q^{-26}+3q^{-28}-4q^{-30}+5q^{-32}-5q^{-34}+5q^{-36}-4q^{-38}+3q^{-40}-3q^{-42}+2q^{-44}-q^{-46}+q^{-48}$
1,0	$q^{34}-q^{30}-q^{28}+q^{26}+2q^{24}-q^{22}-3q^{20}+4q^{16}+3q^{14}-3q^{12}-4q^{10}+2q^{8}+7q^{6}+2q^{4}-5q^{2}-4+3q^{-2}+5q^{-4}+q^{-6}-5q^{-8}-2q^{-10}+3q^{-12}+2q^{-14}-2q^{-16}-2q^{-18}+2q^{-20}+2q^{-22}-2q^{-24}-4q^{-26}+3q^{-30}-4q^{-34}-2q^{-36}+4q^{-38}+4q^{-40}-2q^{-42}-4q^{-44}+q^{-46}+6q^{-48}+3q^{-50}-3q^{-52}-4q^{-54}+4q^{-58}+2q^{-60}-2q^{-62}-3q^{-64}-q^{-66}+2q^{-68}+q^{-70}-q^{-72}-q^{-74}+q^{-78}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{26}-q^{24}+q^{22}-2q^{20}+3q^{18}-4q^{16}+3q^{14}-4q^{12}+6q^{10}-3q^{8}+6q^{6}-q^{4}+6q^{2}+2+2q^{-2}+2q^{-4}-2q^{-6}+2q^{-8}-8q^{-10}+4q^{-12}-10q^{-14}+5q^{-16}-11q^{-18}+6q^{-20}-8q^{-22}+7q^{-24}-5q^{-26}+5q^{-28}-2q^{-30}+4q^{-32}+2q^{-34}+2q^{-38}-2q^{-40}+5q^{-42}-3q^{-44}+3q^{-46}-4q^{-48}+4q^{-50}-3q^{-52}+2q^{-54}-3q^{-56}+2q^{-58}-2q^{-60}+q^{-62}-q^{-64}+q^{-66}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-q^{44}+2q^{42}-3q^{40}+2q^{38}-2q^{36}-q^{34}+6q^{32}-8q^{30}+10q^{28}-9q^{26}+5q^{24}+2q^{22}-10q^{20}+16q^{18}-16q^{16}+14q^{14}-4q^{12}-5q^{10}+15q^{8}-14q^{6}+12q^{4}-4q^{2}-4+10q^{-2}-9q^{-4}+3q^{-6}+6q^{-8}-10q^{-10}+14q^{-12}-9q^{-14}-2q^{-16}+8q^{-18}-16q^{-20}+18q^{-22}-17q^{-24}+7q^{-26}+3q^{-28}-13q^{-30}+18q^{-32}-18q^{-34}+11q^{-36}-3q^{-38}-6q^{-40}+10q^{-42}-12q^{-44}+8q^{-46}-5q^{-50}+6q^{-52}-4q^{-54}-2q^{-56}+7q^{-58}-8q^{-60}+7q^{-62}-4q^{-64}-q^{-66}+6q^{-68}-8q^{-70}+11q^{-72}-8q^{-74}+6q^{-76}-q^{-78}-2q^{-80}+5q^{-82}-8q^{-84}+10q^{-86}-7q^{-88}+4q^{-90}-4q^{-94}+6q^{-96}-6q^{-98}+5q^{-100}-3q^{-102}+q^{-106}-3q^{-108}+2q^{-110}-q^{-112}+q^{-114}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 9"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{4}+3t^{3}-5t^{2}+7t-7+7t^{-1}-5t^{-2}+3t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8-5 z^6-7 z^4-2 z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 39, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{7}-2q^{6}+3q^{5}-5q^{4}+6q^{3}-6q^{2}+6q-4+3q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{8}a^{-2}-7z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-17z^{4}a^{-2}+5z^{4}a^{-4}+5z^{4}-16z^{2}a^{-2}+7z^{2}a^{-4}+7z^{2}-4a^{-2}+2a^{-4}+3$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+2z^{8}a^{-4}+2z^{8}+2az^{7}-2z^{7}a^{-1}-2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-18z^{6}a^{-2}-7z^{6}a^{-4}+2z^{6}a^{-6}-8z^{6}-8az^{5}-2z^{5}a^{-1}-4z^{5}a^{-5}+2z^{5}a^{-7}-4a^{2}z^{4}+31z^{4}a^{-2}+13z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+10z^{4}+7az^{3}+4z^{3}a^{-1}+5z^{3}a^{-3}+4z^{3}a^{-5}-4z^{3}a^{-7}+3a^{2}z^{2}-22z^{2}a^{-2}-8z^{2}a^{-4}+z^{2}a^{-6}-2z^{2}a^{-8}-8z^{2}-az-2za^{-1}-2za^{-3}+za^{-7}+4a^{-2}+2a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 9"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-t^{4}+3t^{3}-5t^{2}+7t-7+7t^{-1}-5t^{-2}+3t^{-3}-t^{-4}$, $q^{7}-2q^{6}+3q^{5}-5q^{4}+6q^{3}-6q^{2}+6q-4+3q^{-1}-2q^{-2}+q^{-3}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-2, -2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-8$ $-16$ $32$ ${\frac {260}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{220}{3}} $128$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1088}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{320}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 144} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{256}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 128} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{2080}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1760}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1231}{15}} ${\frac {3588}{5}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{54844}{45}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2911}{9}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{6271}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 2 is the signature of 10 9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                     1 1 13                   1   -1 11                 2 1   1 9               3 1     -2 7             3 2       1 5           3 3         0 3         3 3           0 1       2 4             2 -1     1 2               -1 -3   1 2                 1 -5   1                   -1 -7 1                     1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-2 q^{19}+4 q^{17}-5 q^{16}+8 q^{14}-10 q^{13}+15 q^{11}-18 q^{10}+22 q^8-23 q^7-q^6+25 q^5-21 q^4-5 q^3+24 q^2-15 q-8+19 q^{-1} -8 q^{-2} -9 q^{-3} +12 q^{-4} -2 q^{-5} -6 q^{-6} +5 q^{-7} -2 q^{-9} + q^{-10} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{39}-2 q^{38}+q^{36}+3 q^{35}-3 q^{34}-3 q^{33}+q^{32}+6 q^{31}-q^{30}-6 q^{29}-2 q^{28}+6 q^{27}+4 q^{26}-5 q^{25}-3 q^{24}+q^{23}+3 q^{22}+5 q^{20}-6 q^{19}-8 q^{18}+5 q^{17}+15 q^{16}-5 q^{15}-21 q^{14}+5 q^{13}+21 q^{12}-24 q^{10}-2 q^9+20 q^8+11 q^7-21 q^6-13 q^5+14 q^4+24 q^3-14 q^2-25 q+6+32 q^{-1} -3 q^{-2} -30 q^{-3} -5 q^{-4} +29 q^{-5} +8 q^{-6} -23 q^{-7} -12 q^{-8} +18 q^{-9} +11 q^{-10} -10 q^{-11} -11 q^{-12} +7 q^{-13} +7 q^{-14} -3 q^{-15} -5 q^{-16} +2 q^{-17} +2 q^{-18} -2 q^{-20} + q^{-21} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{64}-2 q^{63}+q^{61}+5 q^{59}-7 q^{58}-q^{57}+17 q^{54}-13 q^{53}-5 q^{52}-7 q^{51}-3 q^{50}+38 q^{49}-12 q^{48}-7 q^{47}-26 q^{46}-17 q^{45}+64 q^{44}+q^{43}+4 q^{42}-52 q^{41}-52 q^{40}+79 q^{39}+25 q^{38}+43 q^{37}-68 q^{36}-107 q^{35}+68 q^{34}+39 q^{33}+104 q^{32}-54 q^{31}-158 q^{30}+35 q^{29}+29 q^{28}+157 q^{27}-22 q^{26}-179 q^{25}+8 q^{24}+4 q^{23}+178 q^{22}+3 q^{21}-177 q^{20}+q^{19}-15 q^{18}+173 q^{17}+11 q^{16}-161 q^{15}+10 q^{14}-31 q^{13}+151 q^{12}+14 q^{11}-133 q^{10}+25 q^9-46 q^8+111 q^7+13 q^6-92 q^5+50 q^4-57 q^3+59 q^2+q-53+78 q^{-1} -49 q^{-2} +17 q^{-3} -25 q^{-4} -37 q^{-5} +94 q^{-6} -22 q^{-7} +4 q^{-8} -44 q^{-9} -43 q^{-10} +81 q^{-11} +3 q^{-12} +16 q^{-13} -38 q^{-14} -49 q^{-15} +47 q^{-16} +7 q^{-17} +25 q^{-18} -16 q^{-19} -38 q^{-20} +19 q^{-21} +18 q^{-23} -2 q^{-24} -19 q^{-25} +8 q^{-26} -3 q^{-27} +7 q^{-28} + q^{-29} -7 q^{-30} +3 q^{-31} - q^{-32} +2 q^{-33} -2 q^{-35} + q^{-36} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{95}-2 q^{94}+q^{92}+2 q^{90}+q^{89}-5 q^{88}-3 q^{87}+3 q^{86}+2 q^{85}+6 q^{84}+4 q^{83}-11 q^{82}-11 q^{81}+q^{80}+7 q^{79}+15 q^{78}+13 q^{77}-14 q^{76}-27 q^{75}-11 q^{74}+8 q^{73}+31 q^{72}+31 q^{71}-8 q^{70}-43 q^{69}-42 q^{68}-2 q^{67}+50 q^{66}+66 q^{65}+18 q^{64}-58 q^{63}-95 q^{62}-46 q^{61}+60 q^{60}+132 q^{59}+92 q^{58}-52 q^{57}-177 q^{56}-154 q^{55}+25 q^{54}+216 q^{53}+241 q^{52}+24 q^{51}-257 q^{50}-322 q^{49}-96 q^{48}+260 q^{47}+425 q^{46}+181 q^{45}-264 q^{44}-490 q^{43}-268 q^{42}+227 q^{41}+549 q^{40}+350 q^{39}-198 q^{38}-574 q^{37}-406 q^{36}+160 q^{35}+577 q^{34}+446 q^{33}-124 q^{32}-579 q^{31}-462 q^{30}+108 q^{29}+559 q^{28}+465 q^{27}-83 q^{26}-549 q^{25}-466 q^{24}+79 q^{23}+519 q^{22}+458 q^{21}-49 q^{20}-501 q^{19}-448 q^{18}+33 q^{17}+448 q^{16}+439 q^{15}+9 q^{14}-412 q^{13}-412 q^{12}-32 q^{11}+333 q^{10}+381 q^9+79 q^8-278 q^7-335 q^6-83 q^5+189 q^4+273 q^3+110 q^2-136 q-213-81 q^{-1} +76 q^{-2} +137 q^{-3} +71 q^{-4} -53 q^{-5} -89 q^{-6} -20 q^{-7} +41 q^{-8} +41 q^{-9} -7 q^{-10} -53 q^{-11} -30 q^{-12} +43 q^{-13} +66 q^{-14} +28 q^{-15} -42 q^{-16} -87 q^{-17} -44 q^{-18} +42 q^{-19} +83 q^{-20} +58 q^{-21} -14 q^{-22} -77 q^{-23} -68 q^{-24} -3 q^{-25} +52 q^{-26} +64 q^{-27} +23 q^{-28} -31 q^{-29} -51 q^{-30} -28 q^{-31} +9 q^{-32} +36 q^{-33} +28 q^{-34} -3 q^{-35} -18 q^{-36} -17 q^{-37} -8 q^{-38} +10 q^{-39} +14 q^{-40} +2 q^{-41} -5 q^{-42} -2 q^{-43} -5 q^{-44} +6 q^{-46} -3 q^{-48} + q^{-49} - q^{-51} +2 q^{-52} -2 q^{-54} + q^{-55} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{132}-2 q^{131}+q^{129}+2 q^{127}-2 q^{126}+3 q^{125}-7 q^{124}+4 q^{122}+7 q^{120}-5 q^{119}+6 q^{118}-19 q^{117}+8 q^{115}+17 q^{113}-5 q^{112}+14 q^{111}-38 q^{110}-4 q^{109}+6 q^{108}-4 q^{107}+27 q^{106}+4 q^{105}+35 q^{104}-57 q^{103}-2 q^{102}-4 q^{101}-22 q^{100}+23 q^{99}+11 q^{98}+66 q^{97}-68 q^{96}+18 q^{95}-6 q^{94}-43 q^{93}+5 q^{92}+11 q^{91}+85 q^{90}-94 q^{89}+38 q^{88}-4 q^{87}-44 q^{86}+19 q^{85}+48 q^{84}+114 q^{83}-159 q^{82}-16 q^{81}-80 q^{80}-72 q^{79}+95 q^{78}+220 q^{77}+274 q^{76}-187 q^{75}-166 q^{74}-351 q^{73}-284 q^{72}+123 q^{71}+528 q^{70}+686 q^{69}-q^{68}-270 q^{67}-774 q^{66}-771 q^{65}-76 q^{64}+777 q^{63}+1240 q^{62}+447 q^{61}-146 q^{60}-1104 q^{59}-1354 q^{58}-499 q^{57}+787 q^{56}+1649 q^{55}+931 q^{54}+167 q^{53}-1179 q^{52}-1741 q^{51}-904 q^{50}+616 q^{49}+1785 q^{48}+1208 q^{47}+446 q^{46}-1083 q^{45}-1861 q^{44}-1111 q^{43}+453 q^{42}+1757 q^{41}+1275 q^{40}+575 q^{39}-982 q^{38}-1840 q^{37}-1167 q^{36}+362 q^{35}+1696 q^{34}+1263 q^{33}+634 q^{32}-895 q^{31}-1780 q^{30}-1203 q^{29}+249 q^{28}+1602 q^{27}+1251 q^{26}+736 q^{25}-734 q^{24}-1665 q^{23}-1272 q^{22}+20 q^{21}+1397 q^{20}+1216 q^{19}+913 q^{18}-433 q^{17}-1431 q^{16}-1332 q^{15}-319 q^{14}+1037 q^{13}+1080 q^{12}+1084 q^{11}-21 q^{10}-1034 q^9-1273 q^8-648 q^7+560 q^6+774 q^5+1108 q^4+356 q^3-524 q^2-1006 q-784+128 q^{-1} +343 q^{-2} +889 q^{-3} +509 q^{-4} -84 q^{-5} -594 q^{-6} -638 q^{-7} -67 q^{-8} -14 q^{-9} +520 q^{-10} +380 q^{-11} +102 q^{-12} -261 q^{-13} -347 q^{-14} - q^{-15} -124 q^{-16} +244 q^{-17} +149 q^{-18} +51 q^{-19} -161 q^{-20} -165 q^{-21} +123 q^{-22} -44 q^{-23} +178 q^{-24} +51 q^{-25} -20 q^{-26} -199 q^{-27} -161 q^{-28} +114 q^{-29} +12 q^{-30} +194 q^{-31} +92 q^{-32} +27 q^{-33} -179 q^{-34} -186 q^{-35} +17 q^{-36} -36 q^{-37} +140 q^{-38} +118 q^{-39} +104 q^{-40} -79 q^{-41} -130 q^{-42} -28 q^{-43} -85 q^{-44} +42 q^{-45} +69 q^{-46} +107 q^{-47} -6 q^{-48} -50 q^{-49} -7 q^{-50} -70 q^{-51} -9 q^{-52} +13 q^{-53} +60 q^{-54} +8 q^{-55} -12 q^{-56} +14 q^{-57} -32 q^{-58} -12 q^{-59} -5 q^{-60} +23 q^{-61} +2 q^{-62} -5 q^{-63} +12 q^{-64} -9 q^{-65} -4 q^{-66} -4 q^{-67} +8 q^{-68} - q^{-69} -4 q^{-70} +5 q^{-71} -2 q^{-72} - q^{-74} +2 q^{-75} -2 q^{-77} + q^{-78} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{175}-2 q^{174}+q^{172}+2 q^{170}-2 q^{169}+q^{167}-4 q^{166}+q^{165}+2 q^{164}+7 q^{162}-4 q^{161}-5 q^{160}+2 q^{159}-10 q^{158}+4 q^{157}+5 q^{156}+17 q^{154}-3 q^{153}-7 q^{152}-q^{151}-26 q^{150}+7 q^{148}-4 q^{147}+34 q^{146}+10 q^{145}+7 q^{144}+11 q^{143}-49 q^{142}-21 q^{141}-16 q^{140}-28 q^{139}+36 q^{138}+28 q^{137}+52 q^{136}+69 q^{135}-31 q^{134}-37 q^{133}-65 q^{132}-110 q^{131}-17 q^{130}+4 q^{129}+88 q^{128}+178 q^{127}+80 q^{126}+38 q^{125}-79 q^{124}-218 q^{123}-157 q^{122}-140 q^{121}+14 q^{120}+245 q^{119}+251 q^{118}+259 q^{117}+89 q^{116}-200 q^{115}-283 q^{114}-398 q^{113}-272 q^{112}+68 q^{111}+253 q^{110}+500 q^{109}+477 q^{108}+159 q^{107}-89 q^{106}-492 q^{105}-657 q^{104}-470 q^{103}-246 q^{102}+322 q^{101}+741 q^{100}+799 q^{99}+709 q^{98}+73 q^{97}-617 q^{96}-1044 q^{95}-1278 q^{94}-698 q^{93}+240 q^{92}+1129 q^{91}+1837 q^{90}+1462 q^{89}+390 q^{88}-939 q^{87}-2250 q^{86}-2297 q^{85}-1256 q^{84}+516 q^{83}+2479 q^{82}+3033 q^{81}+2154 q^{80}+137 q^{79}-2416 q^{78}-3588 q^{77}-3041 q^{76}-887 q^{75}+2183 q^{74}+3905 q^{73}+3729 q^{72}+1594 q^{71}-1792 q^{70}-3979 q^{69}-4205 q^{68}-2202 q^{67}+1379 q^{66}+3918 q^{65}+4468 q^{64}+2605 q^{63}-1036 q^{62}-3750 q^{61}-4540 q^{60}-2865 q^{59}+768 q^{58}+3591 q^{57}+4546 q^{56}+2971 q^{55}-619 q^{54}-3459 q^{53}-4482 q^{52}-3010 q^{51}+525 q^{50}+3352 q^{49}+4430 q^{48}+3034 q^{47}-463 q^{46}-3278 q^{45}-4374 q^{44}-3049 q^{43}+379 q^{42}+3163 q^{41}+4322 q^{40}+3109 q^{39}-246 q^{38}-3007 q^{37}-4240 q^{36}-3197 q^{35}+22 q^{34}+2773 q^{33}+4124 q^{32}+3300 q^{31}+274 q^{30}-2402 q^{29}-3922 q^{28}-3444 q^{27}-685 q^{26}+1949 q^{25}+3645 q^{24}+3528 q^{23}+1132 q^{22}-1318 q^{21}-3221 q^{20}-3588 q^{19}-1652 q^{18}+631 q^{17}+2687 q^{16}+3507 q^{15}+2095 q^{14}+149 q^{13}-1976 q^{12}-3279 q^{11}-2476 q^{10}-923 q^9+1198 q^8+2888 q^7+2633 q^6+1580 q^5-344 q^4-2271 q^3-2604 q^2-2100 q-444+1600 q^{-1} +2314 q^{-2} +2307 q^{-3} +1094 q^{-4} -829 q^{-5} -1825 q^{-6} -2306 q^{-7} -1525 q^{-8} +196 q^{-9} +1250 q^{-10} +2003 q^{-11} +1657 q^{-12} +318 q^{-13} -652 q^{-14} -1582 q^{-15} -1579 q^{-16} -563 q^{-17} +197 q^{-18} +1087 q^{-19} +1299 q^{-20} +615 q^{-21} +102 q^{-22} -688 q^{-23} -965 q^{-24} -488 q^{-25} -205 q^{-26} +396 q^{-27} +675 q^{-28} +323 q^{-29} +168 q^{-30} -280 q^{-31} -478 q^{-32} -153 q^{-33} -92 q^{-34} +242 q^{-35} +402 q^{-36} +94 q^{-37} +26 q^{-38} -269 q^{-39} -395 q^{-40} -94 q^{-41} -18 q^{-42} +256 q^{-43} +399 q^{-44} +147 q^{-45} +78 q^{-46} -196 q^{-47} -389 q^{-48} -205 q^{-49} -144 q^{-50} +117 q^{-51} +316 q^{-52} +203 q^{-53} +206 q^{-54} + q^{-55} -224 q^{-56} -189 q^{-57} -225 q^{-58} -59 q^{-59} +130 q^{-60} +116 q^{-61} +193 q^{-62} +107 q^{-63} -41 q^{-64} -62 q^{-65} -159 q^{-66} -106 q^{-67} +12 q^{-68} +15 q^{-69} +91 q^{-70} +81 q^{-71} +21 q^{-72} +22 q^{-73} -63 q^{-74} -66 q^{-75} -12 q^{-76} -17 q^{-77} +27 q^{-78} +28 q^{-79} +11 q^{-80} +31 q^{-81} -12 q^{-82} -28 q^{-83} -6 q^{-84} -11 q^{-85} +8 q^{-86} +5 q^{-87} -3 q^{-88} +16 q^{-89} + q^{-90} -8 q^{-91} -2 q^{-92} -4 q^{-93} +4 q^{-94} + q^{-95} -5 q^{-96} +4 q^{-97} +2 q^{-98} -2 q^{-99} - q^{-101} +2 q^{-102} -2 q^{-104} + q^{-105} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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