9 21

9_20

9_22

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 21's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 21 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8
Gauss code	-1, 4, -3, 1, -6, 7, -8, 9, -2, 3, -4, 2, -5, 6, -9, 8, -7, 5
Dowker-Thistlethwaite code	4 10 14 16 12 2 18 8 6
Conway Notation	[31122]

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

[edit Notes on presentations of 9 21]

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["9 21"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 16 12 2 18 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[31122]

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}(5,\{1,1,2,-1,2,-3,2,4,-3,4\})}

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]=

{ 5, 10, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 10.1833 A-Polynomial See Data:9 21/A-polynomial

[edit Notes for 9 21'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 2} Rasmussen s-Invariant 2

[edit Notes for 9 21's four dimensional invariants]

Polynomial invariants

Alexander 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 -2 t^2+11 t-17+11 t^{-1} -2 t^{-2} }
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 -2 z^4+3 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	{ 43, 2 }
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+6q^{5}-7q^{4}+8q^{3}-6q^{2}+5q-3+q^{-1}$
HOMFLY-PT polynomial (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 -z^4 a^{-2} -z^4 a^{-4} +2 z^2 a^{-6} +z^2+ a^{-2} + a^{-6} - a^{-8} }
Kauffman polynomial (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 z^8 a^{-4} +z^8 a^{-6} +3 z^7 a^{-3} +5 z^7 a^{-5} +2 z^7 a^{-7} +4 z^6 a^{-2} +4 z^6 a^{-4} +2 z^6 a^{-6} +2 z^6 a^{-8} +3 z^5 a^{-1} -3 z^5 a^{-3} -10 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-2} -9 z^4 a^{-4} -7 z^4 a^{-6} -5 z^4 a^{-8} +z^4-4 z^3 a^{-1} +2 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-9} +3 z^2 a^{-2} +6 z^2 a^{-4} +5 z^2 a^{-6} +3 z^2 a^{-8} -z^2-z a^{-3} -3 z a^{-5} +2 z a^{-9} - a^{-2} - a^{-6} - a^{-8} }
The A2 invariant	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^4-q^2-1+2 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} + q^{-12} - q^{-14} +2 q^{-16} - q^{-20} + q^{-22} - q^{-24} - q^{-26} }
The G2 invariant	$q^{18}-2q^{16}+4q^{14}-6q^{12}+4q^{10}-q^{8}-4q^{6}+13q^{4}-17q^{2}+22-19q^{-2}+5q^{-4}+10q^{-6}-27q^{-8}+38q^{-10}-39q^{-12}+28q^{-14}-7q^{-16}-17q^{-18}+36q^{-20}-40q^{-22}+30q^{-24}-9q^{-26}-13q^{-28}+24q^{-30}-21q^{-32}+8q^{-34}+18q^{-36}-33q^{-38}+40q^{-40}-24q^{-42}-4q^{-44}+36q^{-46}-58q^{-48}+63q^{-50}-46q^{-52}+17q^{-54}+18q^{-56}-45q^{-58}+57q^{-60}-50q^{-62}+27q^{-64}-25q^{-68}+32q^{-70}-23q^{-72}+7q^{-74}+16q^{-76}-28q^{-78}+26q^{-80}-10q^{-82}-14q^{-84}+36q^{-86}-44q^{-88}+36q^{-90}-17q^{-92}-8q^{-94}+27q^{-96}-37q^{-98}+35q^{-100}-23q^{-102}+6q^{-104}+6q^{-106}-16q^{-108}+16q^{-110}-14q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 9_21.

A1 Invariants.

Weight	Invariant
1	$q^{3}-2q+2q^{-1}-q^{-3}+2q^{-5}+q^{-7}-q^{-9}+2q^{-11}-2q^{-13}+q^{-15}-q^{-17}$
2	$q^{10}-2q^{8}-q^{6}+6q^{4}-4q^{2}-5+11q^{-2}-3q^{-4}-9q^{-6}+11q^{-8}+q^{-10}-8q^{-12}+5q^{-14}+4q^{-16}-2q^{-18}-5q^{-20}+6q^{-22}+5q^{-24}-11q^{-26}+3q^{-28}+8q^{-30}-11q^{-32}-q^{-34}+8q^{-36}-5q^{-38}-2q^{-40}+4q^{-42}-q^{-44}-q^{-46}+q^{-48}$
3	$q^{21}-2q^{19}-q^{17}+3q^{15}+3q^{13}-4q^{11}-8q^{9}+8q^{7}+13q^{5}-9q^{3}-21q+7q^{-1}+32q^{-3}-3q^{-5}-40q^{-7}-5q^{-9}+46q^{-11}+14q^{-13}-41q^{-15}-22q^{-17}+38q^{-19}+26q^{-21}-25q^{-23}-28q^{-25}+13q^{-27}+25q^{-29}+2q^{-31}-21q^{-33}-16q^{-35}+20q^{-37}+26q^{-39}-12q^{-41}-38q^{-43}+6q^{-45}+42q^{-47}+q^{-49}-46q^{-51}-10q^{-53}+41q^{-55}+19q^{-57}-36q^{-59}-24q^{-61}+24q^{-63}+27q^{-65}-13q^{-67}-23q^{-69}+5q^{-71}+17q^{-73}+q^{-75}-11q^{-77}-2q^{-79}+6q^{-81}+2q^{-83}-3q^{-85}-q^{-87}+q^{-89}+q^{-91}-q^{-93}$
4	$q^{36}-2q^{34}-q^{32}+3q^{30}+3q^{26}-7q^{24}-3q^{22}+9q^{20}+q^{18}+9q^{16}-22q^{14}-16q^{12}+22q^{10}+21q^{8}+29q^{6}-50q^{4}-59q^{2}+16+62q^{-2}+97q^{-4}-54q^{-6}-135q^{-8}-51q^{-10}+79q^{-12}+191q^{-14}+6q^{-16}-168q^{-18}-144q^{-20}+28q^{-22}+231q^{-24}+92q^{-26}-118q^{-28}-179q^{-30}-51q^{-32}+173q^{-34}+128q^{-36}-27q^{-38}-136q^{-40}-97q^{-42}+68q^{-44}+109q^{-46}+52q^{-48}-59q^{-50}-104q^{-52}-38q^{-54}+77q^{-56}+114q^{-58}+5q^{-60}-107q^{-62}-126q^{-64}+47q^{-66}+162q^{-68}+69q^{-70}-94q^{-72}-198q^{-74}-2q^{-76}+178q^{-78}+135q^{-80}-42q^{-82}-224q^{-84}-75q^{-86}+123q^{-88}+168q^{-90}+50q^{-92}-170q^{-94}-123q^{-96}+20q^{-98}+125q^{-100}+110q^{-102}-65q^{-104}-96q^{-106}-50q^{-108}+42q^{-110}+91q^{-112}+4q^{-114}-33q^{-116}-47q^{-118}-8q^{-120}+39q^{-122}+13q^{-124}+q^{-126}-19q^{-128}-11q^{-130}+11q^{-132}+3q^{-134}+4q^{-136}-4q^{-138}-4q^{-140}+3q^{-142}+q^{-146}-q^{-148}-q^{-150}+q^{-152}$
5	$q^{55}-2q^{53}-q^{51}+3q^{49}-2q^{41}-2q^{39}+5q^{37}+4q^{35}-6q^{33}-8q^{31}-5q^{29}+6q^{27}+18q^{25}+24q^{23}-3q^{21}-46q^{19}-51q^{17}-10q^{15}+62q^{13}+109q^{11}+65q^{9}-75q^{7}-195q^{5}-154q^{3}+44q+271q^{-1}+307q^{-3}+54q^{-5}-326q^{-7}-489q^{-9}-214q^{-11}+313q^{-13}+652q^{-15}+448q^{-17}-214q^{-19}-771q^{-21}-680q^{-23}+38q^{-25}+782q^{-27}+884q^{-29}+188q^{-31}-703q^{-33}-984q^{-35}-407q^{-37}+529q^{-39}+991q^{-41}+568q^{-43}-320q^{-45}-875q^{-47}-655q^{-49}+101q^{-51}+710q^{-53}+657q^{-55}+70q^{-57}-492q^{-59}-598q^{-61}-214q^{-63}+299q^{-65}+509q^{-67}+303q^{-69}-122q^{-71}-417q^{-73}-373q^{-75}-36q^{-77}+353q^{-79}+446q^{-81}+147q^{-83}-302q^{-85}-520q^{-87}-281q^{-89}+266q^{-91}+622q^{-93}+398q^{-95}-228q^{-97}-701q^{-99}-552q^{-101}+156q^{-103}+774q^{-105}+703q^{-107}-39q^{-109}-782q^{-111}-844q^{-113}-126q^{-115}+722q^{-117}+932q^{-119}+326q^{-121}-568q^{-123}-957q^{-125}-506q^{-127}+348q^{-129}+865q^{-131}+645q^{-133}-88q^{-135}-699q^{-137}-689q^{-139}-133q^{-141}+458q^{-143}+632q^{-145}+303q^{-147}-218q^{-149}-501q^{-151}-364q^{-153}+17q^{-155}+324q^{-157}+340q^{-159}+111q^{-161}-160q^{-163}-263q^{-165}-154q^{-167}+40q^{-169}+161q^{-171}+142q^{-173}+28q^{-175}-80q^{-177}-102q^{-179}-46q^{-181}+26q^{-183}+60q^{-185}+38q^{-187}-q^{-189}-27q^{-191}-27q^{-193}-5q^{-195}+13q^{-197}+12q^{-199}+3q^{-201}-q^{-203}-6q^{-205}-4q^{-207}+3q^{-209}+2q^{-211}-q^{-213}-q^{-219}+q^{-221}+q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}-q^{2}-1+2q^{-2}-q^{-4}+2q^{-6}+q^{-8}+q^{-12}-q^{-14}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
1,1	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 q^{12}-4 q^{10}+10 q^8-20 q^6+34 q^4-52 q^2+78-104 q^{-2} +124 q^{-4} -142 q^{-6} +152 q^{-8} -140 q^{-10} +113 q^{-12} -60 q^{-14} +2 q^{-16} +74 q^{-18} -150 q^{-20} +220 q^{-22} -268 q^{-24} +298 q^{-26} -297 q^{-28} +272 q^{-30} -228 q^{-32} +162 q^{-34} -89 q^{-36} +14 q^{-38} +50 q^{-40} -100 q^{-42} +136 q^{-44} -152 q^{-46} +144 q^{-48} -130 q^{-50} +106 q^{-52} -82 q^{-54} +56 q^{-56} -36 q^{-58} +23 q^{-60} -12 q^{-62} +6 q^{-64} -2 q^{-66} + q^{-68} }
2,0	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 q^{12}-q^{10}-2 q^8+q^6+4 q^4+q^2-7- q^{-2} +8 q^{-4} -7 q^{-8} +2 q^{-10} +8 q^{-12} -4 q^{-16} +2 q^{-18} +4 q^{-20} -3 q^{-22} + q^{-24} +2 q^{-26} -2 q^{-28} +2 q^{-30} +7 q^{-32} - q^{-34} -6 q^{-36} + q^{-38} +5 q^{-40} -4 q^{-42} -9 q^{-44} + q^{-46} +5 q^{-48} - q^{-50} -4 q^{-52} +3 q^{-56} -2 q^{-60} + q^{-64} + q^{-66} }

A3 Invariants.

Weight	Invariant
0,1,0	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 q^8-2 q^6+4 q^2-5+ q^{-2} +8 q^{-4} -8 q^{-6} - q^{-8} +9 q^{-10} -6 q^{-12} -2 q^{-14} +8 q^{-16} +2 q^{-22} +5 q^{-24} - q^{-26} -6 q^{-28} +6 q^{-30} + q^{-32} -10 q^{-34} +6 q^{-36} +2 q^{-38} -9 q^{-40} +3 q^{-42} + q^{-44} -4 q^{-46} +2 q^{-48} + q^{-50} - q^{-52} + q^{-54} }
1,0,0	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 q^5-q^3- q^{-1} +2 q^{-3} - q^{-5} +2 q^{-7} + q^{-9} + q^{-11} + q^{-17} - q^{-19} +2 q^{-21} + q^{-25} - q^{-27} + q^{-29} - q^{-31} - q^{-33} - q^{-35} }

B2 Invariants.

Weight	Invariant
0,1	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 q^8-2 q^6+4 q^4-6 q^2+7-9 q^{-2} +10 q^{-4} -8 q^{-6} +7 q^{-8} -3 q^{-10} +6 q^{-14} -10 q^{-16} +14 q^{-18} -16 q^{-20} +18 q^{-22} -17 q^{-24} +15 q^{-26} -10 q^{-28} +6 q^{-30} - q^{-32} -2 q^{-34} +6 q^{-36} -8 q^{-38} +9 q^{-40} -9 q^{-42} +7 q^{-44} -6 q^{-46} +4 q^{-48} -3 q^{-50} + q^{-52} - q^{-54} }
1,0	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 q^{14}-2 q^{10}-2 q^8+2 q^6+5 q^4-6-4 q^{-2} +6 q^{-4} +9 q^{-6} -2 q^{-8} -10 q^{-10} -4 q^{-12} +8 q^{-14} +7 q^{-16} -4 q^{-18} -8 q^{-20} +2 q^{-22} +8 q^{-24} +2 q^{-26} -6 q^{-28} - q^{-30} +7 q^{-32} +5 q^{-34} -4 q^{-36} -4 q^{-38} +4 q^{-40} +6 q^{-42} -3 q^{-44} -7 q^{-46} + q^{-48} +8 q^{-50} + q^{-52} -9 q^{-54} -7 q^{-56} +6 q^{-58} +9 q^{-60} -2 q^{-62} -10 q^{-64} -4 q^{-66} +6 q^{-68} +5 q^{-70} -2 q^{-72} -5 q^{-74} - q^{-76} +3 q^{-78} +2 q^{-80} - q^{-82} - q^{-84} + q^{-88} }

G2 Invariants.

Weight

Invariant

1,0

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 q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+4 q^{10}-q^8-4 q^6+13 q^4-17 q^2+22-19 q^{-2} +5 q^{-4} +10 q^{-6} -27 q^{-8} +38 q^{-10} -39 q^{-12} +28 q^{-14} -7 q^{-16} -17 q^{-18} +36 q^{-20} -40 q^{-22} +30 q^{-24} -9 q^{-26} -13 q^{-28} +24 q^{-30} -21 q^{-32} +8 q^{-34} +18 q^{-36} -33 q^{-38} +40 q^{-40} -24 q^{-42} -4 q^{-44} +36 q^{-46} -58 q^{-48} +63 q^{-50} -46 q^{-52} +17 q^{-54} +18 q^{-56} -45 q^{-58} +57 q^{-60} -50 q^{-62} +27 q^{-64} -25 q^{-68} +32 q^{-70} -23 q^{-72} +7 q^{-74} +16 q^{-76} -28 q^{-78} +26 q^{-80} -10 q^{-82} -14 q^{-84} +36 q^{-86} -44 q^{-88} +36 q^{-90} -17 q^{-92} -8 q^{-94} +27 q^{-96} -37 q^{-98} +35 q^{-100} -23 q^{-102} +6 q^{-104} +6 q^{-106} -16 q^{-108} +16 q^{-110} -14 q^{-112} +9 q^{-114} -3 q^{-116} -2 q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }

.

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["9 21"];

In[4]:=

Alexander[K][t]

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

Out[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 -2 t^2+11 t-17+11 t^{-1} -2 t^{-2} }

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 -2 z^4+3 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]=

{ 43, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+2q^{7}-4q^{6}+6q^{5}-7q^{4}+8q^{3}-6q^{2}+5q-3+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

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 -z^4 a^{-2} -z^4 a^{-4} +2 z^2 a^{-6} +z^2+ a^{-2} + a^{-6} - a^{-8} }

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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 a^{-4} +z^8 a^{-6} +3 z^7 a^{-3} +5 z^7 a^{-5} +2 z^7 a^{-7} +4 z^6 a^{-2} +4 z^6 a^{-4} +2 z^6 a^{-6} +2 z^6 a^{-8} +3 z^5 a^{-1} -3 z^5 a^{-3} -10 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-2} -9 z^4 a^{-4} -7 z^4 a^{-6} -5 z^4 a^{-8} +z^4-4 z^3 a^{-1} +2 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-9} +3 z^2 a^{-2} +6 z^2 a^{-4} +5 z^2 a^{-6} +3 z^2 a^{-8} -z^2-z a^{-3} -3 z a^{-5} +2 z a^{-9} - a^{-2} - a^{-6} - a^{-8} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, 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\leftrightarrow q^{-1}} ): {K11n129,}

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["9 21"];

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]=

{ $-2t^{2}+11t-17+11t^{-1}-2t^{-2}$, 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^8+2 q^7-4 q^6+6 q^5-7 q^4+8 q^3-6 q^2+5 q-3+ q^{-1} } }

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]=

{K11n129,}

Vassiliev invariants

V2 and V3:

(3, 6)

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 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 12} 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 48} 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 72} 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 238} 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 50} 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 576} 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 1248} $192$ 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 272} 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 288} 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 1152} 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 2856} 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 600} 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{65311}{10}} 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{3182}{5}} ${\frac {18154}{5}}$ 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{235}{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 \frac{5631}{10}}

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 9 21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 1   1 13               3 1   -2 11             3 1     2 9           4 3       -1 7         4 3         1 5       2 4           2 3     3 4             -1 1   1 3               2 -1   2                 -2 -3 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=-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} ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ 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=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}^{3}\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}^{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}^{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=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}^{4}\oplus{\mathbb Z}_2^{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}^{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 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}^{3}\oplus{\mathbb Z}_2^{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}^{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 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}^{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=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^{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=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}\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=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 {\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^{23}-2 q^{22}+6 q^{20}-8 q^{19}-3 q^{18}+19 q^{17}-17 q^{16}-13 q^{15}+38 q^{14}-22 q^{13}-27 q^{12}+54 q^{11}-21 q^{10}-38 q^9+57 q^8-15 q^7-37 q^6+44 q^5-6 q^4-27 q^3+24 q^2-13+8 q^{-1} + q^{-2} -3 q^{-3} + q^{-4} }
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^{45}+2 q^{44}-2 q^{42}-3 q^{41}+7 q^{40}+4 q^{39}-10 q^{38}-12 q^{37}+19 q^{36}+20 q^{35}-22 q^{34}-40 q^{33}+29 q^{32}+60 q^{31}-25 q^{30}-88 q^{29}+17 q^{28}+115 q^{27}-3 q^{26}-139 q^{25}-19 q^{24}+162 q^{23}+38 q^{22}-175 q^{21}-63 q^{20}+188 q^{19}+76 q^{18}-181 q^{17}-99 q^{16}+183 q^{15}+99 q^{14}-158 q^{13}-111 q^{12}+142 q^{11}+102 q^{10}-107 q^9-99 q^8+82 q^7+83 q^6-52 q^5-67 q^4+31 q^3+48 q^2-15 q-32+6 q^{-1} +20 q^{-2} -3 q^{-3} -10 q^{-4} + q^{-5} +4 q^{-6} + q^{-7} -3 q^{-8} + q^{-9} }
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^{74}-2 q^{73}+2 q^{71}-q^{70}+4 q^{69}-9 q^{68}+10 q^{66}-2 q^{65}+12 q^{64}-31 q^{63}-8 q^{62}+30 q^{61}+10 q^{60}+38 q^{59}-78 q^{58}-47 q^{57}+44 q^{56}+47 q^{55}+125 q^{54}-127 q^{53}-139 q^{52}-2 q^{51}+78 q^{50}+300 q^{49}-112 q^{48}-244 q^{47}-145 q^{46}+31 q^{45}+520 q^{44}+6 q^{43}-289 q^{42}-343 q^{41}-118 q^{40}+702 q^{39}+183 q^{38}-246 q^{37}-523 q^{36}-314 q^{35}+806 q^{34}+346 q^{33}-153 q^{32}-638 q^{31}-487 q^{30}+825 q^{29}+458 q^{28}-44 q^{27}-675 q^{26}-602 q^{25}+759 q^{24}+503 q^{23}+67 q^{22}-618 q^{21}-643 q^{20}+594 q^{19}+464 q^{18}+176 q^{17}-463 q^{16}-598 q^{15}+370 q^{14}+336 q^{13}+237 q^{12}-253 q^{11}-459 q^{10}+167 q^9+164 q^8+213 q^7-79 q^6-274 q^5+55 q^4+34 q^3+129 q^2+2 q-123+20 q^{-1} -12 q^{-2} +54 q^{-3} +11 q^{-4} -44 q^{-5} +12 q^{-6} -11 q^{-7} +16 q^{-8} +5 q^{-9} -13 q^{-10} +4 q^{-11} -3 q^{-12} +4 q^{-13} + q^{-14} -3 q^{-15} + q^{-16} }
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^{110}+2 q^{109}-2 q^{107}+q^{106}-2 q^{104}+5 q^{103}+q^{102}-9 q^{101}-q^{100}+5 q^{99}+2 q^{98}+14 q^{97}+2 q^{96}-27 q^{95}-23 q^{94}+5 q^{93}+28 q^{92}+53 q^{91}+24 q^{90}-61 q^{89}-95 q^{88}-51 q^{87}+50 q^{86}+161 q^{85}+138 q^{84}-42 q^{83}-216 q^{82}-245 q^{81}-59 q^{80}+264 q^{79}+409 q^{78}+187 q^{77}-232 q^{76}-552 q^{75}-440 q^{74}+127 q^{73}+692 q^{72}+708 q^{71}+97 q^{70}-726 q^{69}-1031 q^{68}-429 q^{67}+682 q^{66}+1319 q^{65}+830 q^{64}-506 q^{63}-1548 q^{62}-1283 q^{61}+231 q^{60}+1708 q^{59}+1724 q^{58}+100 q^{57}-1758 q^{56}-2131 q^{55}-487 q^{54}+1770 q^{53}+2467 q^{52}+842 q^{51}-1687 q^{50}-2749 q^{49}-1195 q^{48}+1621 q^{47}+2940 q^{46}+1468 q^{45}-1463 q^{44}-3105 q^{43}-1742 q^{42}+1382 q^{41}+3158 q^{40}+1917 q^{39}-1164 q^{38}-3198 q^{37}-2131 q^{36}+1045 q^{35}+3114 q^{34}+2212 q^{33}-739 q^{32}-2992 q^{31}-2341 q^{30}+532 q^{29}+2730 q^{28}+2318 q^{27}-177 q^{26}-2405 q^{25}-2288 q^{24}-77 q^{23}+1974 q^{22}+2098 q^{21}+378 q^{20}-1517 q^{19}-1865 q^{18}-539 q^{17}+1038 q^{16}+1521 q^{15}+659 q^{14}-626 q^{13}-1169 q^{12}-641 q^{11}+294 q^{10}+803 q^9+568 q^8-69 q^7-507 q^6-437 q^5-45 q^4+276 q^3+293 q^2+94 q-127-184 q^{-1} -81 q^{-2} +49 q^{-3} +95 q^{-4} +53 q^{-5} -7 q^{-6} -44 q^{-7} -37 q^{-8} +2 q^{-9} +23 q^{-10} +12 q^{-11} -2 q^{-12} - q^{-13} -10 q^{-14} -4 q^{-15} +11 q^{-16} + q^{-17} -5 q^{-18} + q^{-19} -3 q^{-21} +4 q^{-22} + q^{-23} -3 q^{-24} + q^{-25} }
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^{153}-2 q^{152}+2 q^{150}-q^{149}-2 q^{147}+6 q^{146}-6 q^{145}-2 q^{144}+11 q^{143}-3 q^{142}-4 q^{141}-13 q^{140}+15 q^{139}-12 q^{138}-2 q^{137}+40 q^{136}+5 q^{135}-13 q^{134}-55 q^{133}+11 q^{132}-40 q^{131}+123 q^{129}+70 q^{128}+10 q^{127}-140 q^{126}-54 q^{125}-175 q^{124}-67 q^{123}+257 q^{122}+283 q^{121}+212 q^{120}-147 q^{119}-148 q^{118}-548 q^{117}-432 q^{116}+213 q^{115}+571 q^{114}+754 q^{113}+268 q^{112}+108 q^{111}-1003 q^{110}-1279 q^{109}-499 q^{108}+387 q^{107}+1340 q^{106}+1291 q^{105}+1343 q^{104}-779 q^{103}-2139 q^{102}-2047 q^{101}-1012 q^{100}+990 q^{99}+2286 q^{98}+3647 q^{97}+947 q^{96}-1858 q^{95}-3616 q^{94}-3609 q^{93}-1140 q^{92}+2000 q^{91}+6025 q^{90}+4038 q^{89}+347 q^{88}-3910 q^{87}-6293 q^{86}-4749 q^{85}-209 q^{84}+7174 q^{83}+7295 q^{82}+4009 q^{81}-2419 q^{80}-7843 q^{79}-8568 q^{78}-3712 q^{77}+6702 q^{76}+9565 q^{75}+7832 q^{74}+172 q^{73}-7986 q^{72}-11495 q^{71}-7246 q^{70}+5260 q^{69}+10601 q^{68}+10808 q^{67}+2762 q^{66}-7292 q^{65}-13269 q^{64}-9958 q^{63}+3692 q^{62}+10823 q^{61}+12730 q^{60}+4755 q^{59}-6367 q^{58}-14165 q^{57}-11759 q^{56}+2300 q^{55}+10589 q^{54}+13842 q^{53}+6259 q^{52}-5291 q^{51}-14366 q^{50}-12922 q^{49}+804 q^{48}+9790 q^{47}+14251 q^{46}+7601 q^{45}-3686 q^{44}-13634 q^{43}-13488 q^{42}-1121 q^{41}+7974 q^{40}+13616 q^{39}+8730 q^{38}-1279 q^{37}-11490 q^{36}-13007 q^{35}-3237 q^{34}+4967 q^{33}+11417 q^{32}+9006 q^{31}+1493 q^{30}-7898 q^{29}-10923 q^{28}-4636 q^{27}+1467 q^{26}+7737 q^{25}+7748 q^{24}+3485 q^{23}-3817 q^{22}-7419 q^{21}-4482 q^{20}-1153 q^{19}+3722 q^{18}+5140 q^{17}+3793 q^{16}-754 q^{15}-3722 q^{14}-2957 q^{13}-2008 q^{12}+860 q^{11}+2376 q^{10}+2690 q^9+525 q^8-1187 q^7-1203 q^6-1489 q^5-297 q^4+600 q^3+1306 q^2+522 q-136-164 q^{-1} -670 q^{-2} -356 q^{-3} -44 q^{-4} +448 q^{-5} +191 q^{-6} +44 q^{-7} +128 q^{-8} -191 q^{-9} -149 q^{-10} -105 q^{-11} +126 q^{-12} +19 q^{-13} +3 q^{-14} +97 q^{-15} -35 q^{-16} -35 q^{-17} -46 q^{-18} +42 q^{-19} -11 q^{-20} -13 q^{-21} +34 q^{-22} -7 q^{-23} -4 q^{-24} -14 q^{-25} +18 q^{-26} -4 q^{-27} -9 q^{-28} +9 q^{-29} -3 q^{-30} -3 q^{-32} +4 q^{-33} + q^{-34} -3 q^{-35} + q^{-36} }

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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