9 34

9_33

9_35

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 34's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 34 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X18,11,1,12 X12,17,13,18
Gauss code	1, -4, 3, -7, 6, -1, 2, -3, 5, -6, 8, -9, 7, -5, 4, -2, 9, -8
Dowker-Thistlethwaite code	6 8 10 16 14 18 4 2 12
Conway Notation	[8*20]

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

[edit Notes on presentations of 9 34]

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 34"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X18,11,1,12 X12,17,13,18

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 8 10 16 14 18 4 2 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[8*20]

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

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]=

{ 4, 9, 4 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index 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 \{4,6\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 14.3446 A-Polynomial See Data:9 34/A-polynomial

[edit Notes for 9 34'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 3} Rasmussen s-Invariant 0

[edit Notes for 9 34'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 -t^3+6 t^2-16 t+23-16 t^{-1} +6 t^{-2} - t^{-3} }
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^6-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	{ 69, 0 }
Jones 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 q^4-4 q^3+8 q^2-10 q+12-12 q^{-1} +10 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5} }
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+3a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+a^{2}+a^{-2}-1$
Kauffman polynomial (db, data sources)	$3a^{2}z^{8}+3z^{8}+6a^{3}z^{7}+14az^{7}+8z^{7}a^{-1}+4a^{4}z^{6}+5a^{2}z^{6}+8z^{6}a^{-2}+9z^{6}+a^{5}z^{5}-11a^{3}z^{5}-26az^{5}-10z^{5}a^{-1}+4z^{5}a^{-3}-7a^{4}z^{4}-19a^{2}z^{4}-10z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}-a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}+3a^{4}z^{2}+10a^{2}z^{2}+4z^{2}a^{-2}+11z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$
The A2 invariant	$-q^{16}+q^{14}+2q^{12}-2q^{10}+2q^{8}-q^{6}-q^{4}+2q^{2}-2+3q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+14q^{72}-12q^{70}-q^{68}+27q^{66}-55q^{64}+83q^{62}-84q^{60}+44q^{58}+24q^{56}-112q^{54}+181q^{52}-188q^{50}+127q^{48}-11q^{46}-114q^{44}+205q^{42}-213q^{40}+135q^{38}-7q^{36}-116q^{34}+167q^{32}-131q^{30}+24q^{28}+103q^{26}-178q^{24}+183q^{22}-102q^{20}-37q^{18}+174q^{16}-269q^{14}+272q^{12}-182q^{10}+32q^{8}+130q^{6}-244q^{4}+280q^{2}-217+85q^{-2}+58q^{-4}-170q^{-6}+191q^{-8}-117q^{-10}-6q^{-12}+123q^{-14}-165q^{-16}+125q^{-18}-16q^{-20}-113q^{-22}+195q^{-24}-206q^{-26}+141q^{-28}-26q^{-30}-88q^{-32}+159q^{-34}-165q^{-36}+126q^{-38}-55q^{-40}-10q^{-42}+48q^{-44}-68q^{-46}+60q^{-48}-38q^{-50}+18q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_34.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+3q^{9}-3q^{7}+3q^{5}-2q^{3}+2q^{-1}-2q^{-3}+4q^{-5}-3q^{-7}+q^{-9}$
2	$q^{32}-3q^{30}-q^{28}+12q^{26}-8q^{24}-16q^{22}+25q^{20}+q^{18}-32q^{16}+22q^{14}+16q^{12}-28q^{10}+6q^{8}+19q^{6}-10q^{4}-13q^{2}+11+14q^{-2}-25q^{-4}-2q^{-6}+33q^{-8}-21q^{-10}-15q^{-12}+30q^{-14}-7q^{-16}-15q^{-18}+10q^{-20}+q^{-22}-3q^{-24}+q^{-26}$
3	$-q^{63}+3q^{61}+q^{59}-8q^{57}-7q^{55}+16q^{53}+30q^{51}-24q^{49}-64q^{47}+6q^{45}+107q^{43}+40q^{41}-138q^{39}-106q^{37}+135q^{35}+178q^{33}-96q^{31}-232q^{29}+37q^{27}+250q^{25}+28q^{23}-234q^{21}-86q^{19}+198q^{17}+119q^{15}-144q^{13}-137q^{11}+85q^{9}+150q^{7}-25q^{5}-151q^{3}-42q+150q^{-1}+110q^{-3}-132q^{-5}-180q^{-7}+98q^{-9}+234q^{-11}-44q^{-13}-255q^{-15}-22q^{-17}+239q^{-19}+81q^{-21}-187q^{-23}-113q^{-25}+121q^{-27}+105q^{-29}-53q^{-31}-82q^{-33}+15q^{-35}+48q^{-37}-2q^{-39}-18q^{-41}-3q^{-43}+7q^{-45}+q^{-47}-3q^{-49}+q^{-51}$
4	$q^{104}-3q^{102}-q^{100}+8q^{98}+3q^{96}-q^{94}-30q^{92}-20q^{90}+45q^{88}+67q^{86}+55q^{84}-123q^{82}-211q^{80}-29q^{78}+231q^{76}+437q^{74}+26q^{72}-539q^{70}-606q^{68}-39q^{66}+965q^{64}+866q^{62}-237q^{60}-1287q^{58}-1138q^{56}+669q^{54}+1720q^{52}+979q^{50}-1012q^{48}-2141q^{46}-546q^{44}+1536q^{42}+2020q^{40}+111q^{38}-2061q^{36}-1539q^{34}+565q^{32}+2042q^{30}+996q^{28}-1229q^{26}-1668q^{24}-264q^{22}+1418q^{20}+1232q^{18}-399q^{16}-1348q^{14}-721q^{12}+758q^{10}+1234q^{8}+313q^{6}-1013q^{4}-1169q^{2}+46+1278q^{-2}+1203q^{-4}-513q^{-6}-1679q^{-8}-981q^{-10}+1003q^{-12}+2097q^{-14}+482q^{-16}-1631q^{-18}-2013q^{-20}+11q^{-22}+2191q^{-24}+1535q^{-26}-628q^{-28}-2103q^{-30}-1083q^{-32}+1160q^{-34}+1627q^{-36}+495q^{-38}-1100q^{-40}-1210q^{-42}+49q^{-44}+779q^{-46}+690q^{-48}-151q^{-50}-563q^{-52}-227q^{-54}+109q^{-56}+291q^{-58}+71q^{-60}-111q^{-62}-74q^{-64}-24q^{-66}+53q^{-68}+22q^{-70}-13q^{-72}-6q^{-74}-6q^{-76}+7q^{-78}+q^{-80}-3q^{-82}+q^{-84}$
5	$-q^{155}+3q^{153}+q^{151}-8q^{149}-3q^{147}+5q^{145}+15q^{143}+20q^{141}-q^{139}-59q^{137}-86q^{135}-15q^{133}+123q^{131}+240q^{129}+177q^{127}-129q^{125}-545q^{123}-631q^{121}-100q^{119}+802q^{117}+1419q^{115}+982q^{113}-623q^{111}-2384q^{109}-2611q^{107}-542q^{105}+2761q^{103}+4754q^{101}+3135q^{99}-1778q^{97}-6538q^{95}-6794q^{93}-1213q^{91}+6706q^{89}+10559q^{87}+6079q^{85}-4446q^{83}-12971q^{81}-11708q^{79}-320q^{77}+12798q^{75}+16578q^{73}+6682q^{71}-9776q^{69}-19250q^{67}-13013q^{65}+4536q^{63}+19015q^{61}+17848q^{59}+1556q^{57}-16212q^{55}-20243q^{53}-7017q^{51}+11839q^{49}+20042q^{47}+10881q^{45}-7083q^{43}-17954q^{41}-12766q^{39}+2989q^{37}+14851q^{35}+12931q^{33}+56q^{31}-11650q^{29}-12114q^{27}-1995q^{25}+8933q^{23}+11006q^{21}+3244q^{19}-6807q^{17}-10249q^{15}-4442q^{13}+5184q^{11}+10107q^{9}+6046q^{7}-3522q^{5}-10451q^{3}-8507q+1299q^{-1}+10908q^{-3}+11682q^{-5}+1980q^{-7}-10674q^{-9}-15162q^{-11}-6449q^{-13}+9048q^{-15}+18039q^{-17}+11723q^{-19}-5574q^{-21}-19230q^{-23}-16847q^{-25}+368q^{-27}+17919q^{-29}+20551q^{-31}+5661q^{-33}-13978q^{-35}-21629q^{-37}-11107q^{-39}+8117q^{-41}+19635q^{-43}+14545q^{-45}-1807q^{-47}-15140q^{-49}-15083q^{-51}-3294q^{-53}+9385q^{-55}+12979q^{-57}+6171q^{-59}-4122q^{-61}-9312q^{-63}-6523q^{-65}+394q^{-67}+5411q^{-69}+5270q^{-71}+1385q^{-73}-2442q^{-75}-3377q^{-77}-1635q^{-79}+677q^{-81}+1733q^{-83}+1211q^{-85}+37q^{-87}-739q^{-89}-650q^{-91}-151q^{-93}+224q^{-95}+283q^{-97}+113q^{-99}-60q^{-101}-110q^{-103}-41q^{-105}+20q^{-107}+27q^{-109}+11q^{-111}-q^{-113}-9q^{-115}-6q^{-117}+7q^{-119}+q^{-121}-3q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+2q^{12}-2q^{10}+2q^{8}-q^{6}-q^{4}+2q^{2}-2+3q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}$
1,1	$q^{44}-6q^{42}+20q^{40}-50q^{38}+107q^{36}-204q^{34}+344q^{32}-514q^{30}+701q^{28}-864q^{26}+950q^{24}-938q^{22}+791q^{20}-502q^{18}+98q^{16}+374q^{14}-840q^{12}+1276q^{10}-1606q^{8}+1788q^{6}-1806q^{4}+1650q^{2}-1352+934q^{-2}-456q^{-4}-16q^{-6}+432q^{-8}-726q^{-10}+888q^{-12}-918q^{-14}+850q^{-16}-712q^{-18}+538q^{-20}-376q^{-22}+248q^{-24}-148q^{-26}+77q^{-28}-38q^{-30}+18q^{-32}-6q^{-34}+q^{-36}$
2,0	$q^{42}-q^{40}-3q^{38}+7q^{34}+5q^{32}-10q^{30}-7q^{28}+9q^{26}+7q^{24}-13q^{22}-7q^{20}+13q^{18}+8q^{16}-11q^{14}-q^{12}+13q^{10}-5q^{8}-3q^{6}+4q^{4}-q^{2}-7+5q^{-2}+7q^{-4}-13q^{-6}-4q^{-8}+16q^{-10}+6q^{-12}-17q^{-14}+q^{-16}+14q^{-18}-9q^{-22}-3q^{-24}+6q^{-26}-2q^{-30}+q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-3q^{32}+q^{30}+6q^{28}-12q^{26}+6q^{24}+13q^{22}-22q^{20}+10q^{18}+15q^{16}-24q^{14}+7q^{12}+14q^{10}-13q^{8}-2q^{6}+7q^{4}+3q^{2}-6-6q^{-2}+18q^{-4}-6q^{-6}-17q^{-8}+25q^{-10}-5q^{-12}-18q^{-14}+20q^{-16}-3q^{-18}-10q^{-20}+9q^{-22}-3q^{-26}+q^{-28}$
1,0,0	$-q^{21}+q^{19}+2q^{15}-2q^{13}+3q^{11}-2q^{9}+q^{7}-q^{5}+q^{3}-q^{-1}+2q^{-3}-2q^{-5}+2q^{-7}-q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}-q^{42}-3q^{40}+2q^{38}+4q^{36}-4q^{34}-5q^{32}+9q^{30}+7q^{28}-14q^{26}-4q^{24}+20q^{22}-q^{20}-20q^{18}+9q^{16}+17q^{14}-14q^{12}-12q^{10}+16q^{8}+2q^{6}-19q^{4}+10q^{2}+18-16q^{-2}-5q^{-4}+24q^{-6}-5q^{-8}-21q^{-10}+8q^{-12}+15q^{-14}-12q^{-16}-10q^{-18}+14q^{-20}+7q^{-22}-10q^{-24}-q^{-26}+7q^{-28}-2q^{-30}-2q^{-32}+q^{-34}$
1,0,0,0	$-q^{26}+q^{24}+2q^{18}-2q^{16}+3q^{14}-q^{12}+q^{8}-q^{6}+q^{4}-q^{2}+1-2q^{-2}+2q^{-4}-2q^{-6}+2q^{-8}+3q^{-14}-2q^{-16}+q^{-18}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+3q^{32}-7q^{30}+12q^{28}-18q^{26}+24q^{24}-25q^{22}+26q^{20}-22q^{18}+15q^{16}-4q^{14}-9q^{12}+22q^{10}-35q^{8}+44q^{6}-49q^{4}+49q^{2}-44+34q^{-2}-20q^{-4}+8q^{-6}+5q^{-8}-15q^{-10}+23q^{-12}-26q^{-14}+26q^{-16}-23q^{-18}+18q^{-20}-11q^{-22}+6q^{-24}-3q^{-26}+q^{-28}$
1,0	$q^{56}-3q^{52}-3q^{50}+4q^{48}+9q^{46}-q^{44}-15q^{42}-8q^{40}+18q^{38}+19q^{36}-10q^{34}-27q^{32}-3q^{30}+28q^{28}+15q^{26}-21q^{24}-22q^{22}+10q^{20}+23q^{18}-q^{16}-20q^{14}-3q^{12}+18q^{10}+7q^{8}-16q^{6}-11q^{4}+13q^{2}+16-10q^{-2}-20q^{-4}+7q^{-6}+23q^{-8}+q^{-10}-25q^{-12}-10q^{-14}+23q^{-16}+22q^{-18}-14q^{-20}-27q^{-22}+q^{-24}+23q^{-26}+10q^{-28}-13q^{-30}-14q^{-32}+3q^{-34}+10q^{-36}+3q^{-38}-3q^{-40}-3q^{-42}+q^{-46}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-3q^{44}+4q^{42}-6q^{40}+10q^{38}-15q^{36}+17q^{34}-18q^{32}+22q^{30}-21q^{28}+18q^{26}-15q^{24}+14q^{22}-5q^{20}-5q^{18}+10q^{16}-14q^{14}+25q^{12}-33q^{10}+33q^{8}-35q^{6}+40q^{4}-37q^{2}+31-29q^{-2}+25q^{-4}-12q^{-6}+4q^{-8}-2q^{-10}-5q^{-12}+16q^{-14}-17q^{-16}+18q^{-18}-22q^{-20}+23q^{-22}-16q^{-24}+14q^{-26}-14q^{-28}+10q^{-30}-4q^{-32}+3q^{-34}-3q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-3q^{78}+7q^{76}-13q^{74}+14q^{72}-12q^{70}-q^{68}+27q^{66}-55q^{64}+83q^{62}-84q^{60}+44q^{58}+24q^{56}-112q^{54}+181q^{52}-188q^{50}+127q^{48}-11q^{46}-114q^{44}+205q^{42}-213q^{40}+135q^{38}-7q^{36}-116q^{34}+167q^{32}-131q^{30}+24q^{28}+103q^{26}-178q^{24}+183q^{22}-102q^{20}-37q^{18}+174q^{16}-269q^{14}+272q^{12}-182q^{10}+32q^{8}+130q^{6}-244q^{4}+280q^{2}-217+85q^{-2}+58q^{-4}-170q^{-6}+191q^{-8}-117q^{-10}-6q^{-12}+123q^{-14}-165q^{-16}+125q^{-18}-16q^{-20}-113q^{-22}+195q^{-24}-206q^{-26}+141q^{-28}-26q^{-30}-88q^{-32}+159q^{-34}-165q^{-36}+126q^{-38}-55q^{-40}-10q^{-42}+48q^{-44}-68q^{-46}+60q^{-48}-38q^{-50}+18q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

.

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 34"];

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 -t^3+6 t^2-16 t+23-16 t^{-1} +6 t^{-2} - t^{-3} }

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^6-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]=

{ 69, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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^4-4 q^3+8 q^2-10 q+12-12 q^{-1} +10 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+3a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+a^{2}+a^{-2}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{2}z^{8}+3z^{8}+6a^{3}z^{7}+14az^{7}+8z^{7}a^{-1}+4a^{4}z^{6}+5a^{2}z^{6}+8z^{6}a^{-2}+9z^{6}+a^{5}z^{5}-11a^{3}z^{5}-26az^{5}-10z^{5}a^{-1}+4z^{5}a^{-3}-7a^{4}z^{4}-19a^{2}z^{4}-10z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}-a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}+3a^{4}z^{2}+10a^{2}z^{2}+4z^{2}a^{-2}+11z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n32, K11n119,}

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["9 34"];

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]=

{ $-t^{3}+6t^{2}-16t+23-16t^{-1}+6t^{-2}-t^{-3}$, $q^{4}-4q^{3}+8q^{2}-10q+12-12q^{-1}+10q^{-2}-7q^{-3}+4q^{-4}-q^{-5}$ }

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]=

{K11n32, K11n119,}

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:

(-1, 0)

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 $-4$ $0$ $8$ ${\frac {34}{3}}$ ${\frac {14}{3}}$ $0$ $0$ $32$ $-32$ $-{\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ $-{\frac {56}{3}}$ $-{\frac {1231}{30}}$ $-{\frac {338}{15}}$ $-{\frac {1742}{45}}$ ${\frac {655}{18}}$ $-{\frac {271}{30}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (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=} 0 is the signature of 9 34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               5 1   4 3             5 3     -2 1           7 5       2 -1         6 6         0 -3       4 6           -2 -5     3 6             3 -7   1 4               -3 -9   3                 3 -11 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=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 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}} 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} 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} 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^{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}^{6}\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=-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}^{6}\oplus{\mathbb Z}_2^{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}^{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 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}^{6}\oplus{\mathbb Z}_2^{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}^{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 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}^{5}\oplus{\mathbb Z}_2^{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}^{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 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^{5}} ${\mathbb {Z} }^{5}$ 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 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}\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}_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^{12}-4 q^{11}+4 q^{10}+10 q^9-29 q^8+12 q^7+47 q^6-74 q^5+6 q^4+101 q^3-109 q^2-17 q+140-112 q^{-1} -41 q^{-2} +143 q^{-3} -83 q^{-4} -54 q^{-5} +109 q^{-6} -39 q^{-7} -48 q^{-8} +55 q^{-9} -6 q^{-10} -24 q^{-11} +14 q^{-12} +2 q^{-13} -4 q^{-14} + q^{-15} }
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^{24}-4 q^{23}+4 q^{22}+6 q^{21}-9 q^{20}-19 q^{19}+20 q^{18}+56 q^{17}-42 q^{16}-116 q^{15}+49 q^{14}+214 q^{13}-26 q^{12}-350 q^{11}-25 q^{10}+482 q^9+132 q^8-611 q^7-258 q^6+693 q^5+410 q^4-747 q^3-536 q^2+741 q+652-707 q^{-1} -728 q^{-2} +632 q^{-3} +778 q^{-4} -532 q^{-5} -793 q^{-6} +410 q^{-7} +771 q^{-8} -269 q^{-9} -714 q^{-10} +126 q^{-11} +623 q^{-12} -7 q^{-13} -492 q^{-14} -87 q^{-15} +354 q^{-16} +129 q^{-17} -218 q^{-18} -130 q^{-19} +113 q^{-20} +97 q^{-21} -40 q^{-22} -63 q^{-23} +12 q^{-24} +27 q^{-25} -9 q^{-27} -2 q^{-28} +4 q^{-29} - q^{-30} }
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^{40}-4 q^{39}+4 q^{38}+6 q^{37}-13 q^{36}+q^{35}-11 q^{34}+39 q^{33}+37 q^{32}-90 q^{31}-49 q^{30}-48 q^{29}+221 q^{28}+257 q^{27}-272 q^{26}-385 q^{25}-384 q^{24}+633 q^{23}+1098 q^{22}-183 q^{21}-1115 q^{20}-1643 q^{19}+743 q^{18}+2693 q^{17}+949 q^{16}-1582 q^{15}-3886 q^{14}-277 q^{13}+4168 q^{12}+3112 q^{11}-926 q^{10}-6066 q^9-2301 q^8+4550 q^7+5225 q^6+689 q^5-7160 q^4-4285 q^3+3852 q^2+6391 q+2405-7085 q^{-1} -5517 q^{-2} +2637 q^{-3} +6547 q^{-4} +3731 q^{-5} -6164 q^{-6} -5993 q^{-7} +1158 q^{-8} +5920 q^{-9} +4680 q^{-10} -4533 q^{-11} -5807 q^{-12} -524 q^{-13} +4516 q^{-14} +5119 q^{-15} -2308 q^{-16} -4761 q^{-17} -2001 q^{-18} +2412 q^{-19} +4597 q^{-20} -136 q^{-21} -2852 q^{-22} -2485 q^{-23} +330 q^{-24} +3002 q^{-25} +993 q^{-26} -861 q^{-27} -1744 q^{-28} -721 q^{-29} +1195 q^{-30} +844 q^{-31} +189 q^{-32} -641 q^{-33} -622 q^{-34} +191 q^{-35} +277 q^{-36} +256 q^{-37} -76 q^{-38} -211 q^{-39} -15 q^{-40} +17 q^{-41} +74 q^{-42} +12 q^{-43} -33 q^{-44} -3 q^{-45} -5 q^{-46} +9 q^{-47} +2 q^{-48} -4 q^{-49} + q^{-50} }
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^{60}-4 q^{59}+4 q^{58}+6 q^{57}-13 q^{56}-3 q^{55}+9 q^{54}+8 q^{53}+20 q^{52}-q^{51}-74 q^{50}-72 q^{49}+59 q^{48}+181 q^{47}+190 q^{46}-60 q^{45}-449 q^{44}-571 q^{43}-30 q^{42}+957 q^{41}+1364 q^{40}+462 q^{39}-1505 q^{38}-2883 q^{37}-1772 q^{36}+1892 q^{35}+5191 q^{34}+4347 q^{33}-1364 q^{32}-7900 q^{31}-8689 q^{30}-897 q^{29}+10381 q^{28}+14640 q^{27}+5444 q^{26}-11494 q^{25}-21368 q^{24}-12686 q^{23}+10324 q^{22}+27973 q^{21}+21796 q^{20}-6404 q^{19}-32886 q^{18}-31910 q^{17}-198 q^{16}+35624 q^{15}+41435 q^{14}+8486 q^{13}-35518 q^{12}-49461 q^{11}-17413 q^{10}+33241 q^9+55091 q^8+25783 q^7-29202 q^6-58452 q^5-32910 q^4+24528 q^3+59579 q^2+38437 q-19500-59226 q^{-1} -42519 q^{-2} +14722 q^{-3} +57635 q^{-4} +45366 q^{-5} -9932 q^{-6} -55165 q^{-7} -47442 q^{-8} +5096 q^{-9} +51828 q^{-10} +48808 q^{-11} +119 q^{-12} -47403 q^{-13} -49515 q^{-14} -5832 q^{-15} +41709 q^{-16} +49272 q^{-17} +11825 q^{-18} -34528 q^{-19} -47595 q^{-20} -17694 q^{-21} +25954 q^{-22} +44084 q^{-23} +22696 q^{-24} -16564 q^{-25} -38434 q^{-26} -25897 q^{-27} +7098 q^{-28} +30858 q^{-29} +26727 q^{-30} +1204 q^{-31} -22142 q^{-32} -24730 q^{-33} -7381 q^{-34} +13309 q^{-35} +20490 q^{-36} +10678 q^{-37} -5684 q^{-38} -14834 q^{-39} -11161 q^{-40} +191 q^{-41} +9102 q^{-42} +9415 q^{-43} +2841 q^{-44} -4309 q^{-45} -6681 q^{-46} -3662 q^{-47} +1183 q^{-48} +3834 q^{-49} +3097 q^{-50} +451 q^{-51} -1768 q^{-52} -2043 q^{-53} -810 q^{-54} +531 q^{-55} +1028 q^{-56} +678 q^{-57} -7 q^{-58} -438 q^{-59} -373 q^{-60} -86 q^{-61} +126 q^{-62} +147 q^{-63} +79 q^{-64} -22 q^{-65} -67 q^{-66} -23 q^{-67} +9 q^{-68} +9 q^{-69} +8 q^{-70} +5 q^{-71} -9 q^{-72} -2 q^{-73} +4 q^{-74} - q^{-75} }
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^{84}-4 q^{83}+4 q^{82}+6 q^{81}-13 q^{80}-3 q^{79}+5 q^{78}+28 q^{77}-11 q^{76}-18 q^{75}+15 q^{74}-87 q^{73}-16 q^{72}+93 q^{71}+214 q^{70}+60 q^{69}-172 q^{68}-226 q^{67}-596 q^{66}-201 q^{65}+616 q^{64}+1477 q^{63}+1095 q^{62}-260 q^{61}-1676 q^{60}-3767 q^{59}-2705 q^{58}+1083 q^{57}+6329 q^{56}+7879 q^{55}+4218 q^{54}-3277 q^{53}-14376 q^{52}-16750 q^{51}-7389 q^{50}+12518 q^{49}+28184 q^{48}+29057 q^{47}+10529 q^{46}-27268 q^{45}-53389 q^{44}-48973 q^{43}-4566 q^{42}+51710 q^{41}+87565 q^{40}+72590 q^{39}-6610 q^{38}-94578 q^{37}-136646 q^{36}-85215 q^{35}+30912 q^{34}+151409 q^{33}+191466 q^{32}+92051 q^{31}-82084 q^{30}-229420 q^{29}-229647 q^{28}-78266 q^{27}+154170 q^{26}+312752 q^{25}+257079 q^{24}+23502 q^{23}-255646 q^{22}-371665 q^{21}-251033 q^{20}+63988 q^{19}+363413 q^{18}+413536 q^{17}+186123 q^{16}-191750 q^{15}-441171 q^{14}-408894 q^{13}-75882 q^{12}+328123 q^{11}+497760 q^{10}+330085 q^9-83203 q^8-430541 q^7-498059 q^6-197997 q^5+250860 q^4+508808 q^3+414062 q^2+14014 q-380135-523707 q^{-1} -273997 q^{-2} +175604 q^{-3} +482484 q^{-4} +448797 q^{-5} +83170 q^{-6} -323134 q^{-7} -517342 q^{-8} -318813 q^{-9} +109774 q^{-10} +442844 q^{-11} +462028 q^{-12} +141818 q^{-13} -259898 q^{-14} -495577 q^{-15} -356319 q^{-16} +34442 q^{-17} +384479 q^{-18} +463670 q^{-19} +209578 q^{-20} -170599 q^{-21} -447370 q^{-22} -389077 q^{-23} -65372 q^{-24} +286036 q^{-25} +435588 q^{-26} +280278 q^{-27} -44606 q^{-28} -348258 q^{-29} -389585 q^{-30} -173593 q^{-31} +140530 q^{-32} +348024 q^{-33} +315023 q^{-34} +92020 q^{-35} -193995 q^{-36} -321541 q^{-37} -239573 q^{-38} -16250 q^{-39} +199049 q^{-40} +271427 q^{-41} +178557 q^{-42} -29574 q^{-43} -186479 q^{-44} -217687 q^{-45} -115811 q^{-46} +42172 q^{-47} +156660 q^{-48} +170542 q^{-49} +71966 q^{-50} -45861 q^{-51} -123612 q^{-52} -118831 q^{-53} -48992 q^{-54} +38852 q^{-55} +94087 q^{-56} +80016 q^{-57} +30049 q^{-58} -29789 q^{-59} -61282 q^{-60} -55241 q^{-61} -18946 q^{-62} +22114 q^{-63} +38216 q^{-64} +33982 q^{-65} +11198 q^{-66} -11458 q^{-67} -24328 q^{-68} -20110 q^{-69} -5535 q^{-70} +5890 q^{-71} +13020 q^{-72} +10756 q^{-73} +4175 q^{-74} -3742 q^{-75} -6640 q^{-76} -4888 q^{-77} -2203 q^{-78} +1492 q^{-79} +2913 q^{-80} +2695 q^{-81} +623 q^{-82} -680 q^{-83} -1006 q^{-84} -1063 q^{-85} -320 q^{-86} +212 q^{-87} +552 q^{-88} +242 q^{-89} +59 q^{-90} -27 q^{-91} -163 q^{-92} -92 q^{-93} -26 q^{-94} +72 q^{-95} +16 q^{-96} +2 q^{-97} +15 q^{-98} -14 q^{-99} -8 q^{-100} -5 q^{-101} +9 q^{-102} +2 q^{-103} -4 q^{-104} + q^{-105} }
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^{112}-4 q^{111}+4 q^{110}+6 q^{109}-13 q^{108}-3 q^{107}+5 q^{106}+24 q^{105}+9 q^{104}-49 q^{103}-2 q^{102}+2 q^{101}-31 q^{100}+28 q^{99}+75 q^{98}+158 q^{97}+48 q^{96}-291 q^{95}-318 q^{94}-260 q^{93}-66 q^{92}+534 q^{91}+917 q^{90}+1208 q^{89}+482 q^{88}-1471 q^{87}-2844 q^{86}-3329 q^{85}-1711 q^{84}+2371 q^{83}+6470 q^{82}+9138 q^{81}+6836 q^{80}-2254 q^{79}-13194 q^{78}-21631 q^{77}-19779 q^{76}-3481 q^{75}+20489 q^{74}+43605 q^{73}+49176 q^{72}+25233 q^{71}-21814 q^{70}-75414 q^{69}-103193 q^{68}-77368 q^{67}+217 q^{66}+106234 q^{65}+185323 q^{64}+179387 q^{63}+71464 q^{62}-113375 q^{61}-286642 q^{60}-341928 q^{59}-223202 q^{58}+56051 q^{57}+375169 q^{56}+559280 q^{55}+480840 q^{54}+112560 q^{53}-400811 q^{52}-797484 q^{51}-843686 q^{50}-430400 q^{49}+298839 q^{48}+993820 q^{47}+1278907 q^{46}+910274 q^{45}-15765 q^{44}-1073367 q^{43}-1718343 q^{42}-1520174 q^{41}-470669 q^{40}+964613 q^{39}+2073346 q^{38}+2192067 q^{37}+1139954 q^{36}-633602 q^{35}-2263880 q^{34}-2829249 q^{33}-1922465 q^{32}+89307 q^{31}+2235800 q^{30}+3341119 q^{29}+2725638 q^{28}+609765 q^{27}-1986923 q^{26}-3661296 q^{25}-3449927 q^{24}-1377286 q^{23}+1555703 q^{22}+3768099 q^{21}+4023424 q^{20}+2119469 q^{19}-1015128 q^{18}-3681754 q^{17}-4408420 q^{16}-2762212 q^{15}+444233 q^{14}+3454987 q^{13}+4609291 q^{12}+3261430 q^{11}+87493 q^{10}-3149968 q^9-4658362 q^8-3610169 q^7-537899 q^6+2826220 q^5+4604709 q^4+3825894 q^3+890834 q^2-2522244 q-4494135-3945409 q^{-1} -1157376 q^{-2} +2257688 q^{-3} +4364984 q^{-4} +4006145 q^{-5} +1360966 q^{-6} -2029292 q^{-7} -4235883 q^{-8} -4042274 q^{-9} -1535296 q^{-10} +1820136 q^{-11} +4112104 q^{-12} +4076227 q^{-13} +1709872 q^{-14} -1604177 q^{-15} -3981713 q^{-16} -4116003 q^{-17} -1909743 q^{-18} +1351280 q^{-19} +3823412 q^{-20} +4156866 q^{-21} +2147420 q^{-22} -1036621 q^{-23} -3607912 q^{-24} -4177817 q^{-25} -2420560 q^{-26} +641548 q^{-27} +3304202 q^{-28} +4148097 q^{-29} +2710187 q^{-30} -164342 q^{-31} -2887912 q^{-32} -4027779 q^{-33} -2978499 q^{-34} -377124 q^{-35} +2346613 q^{-36} +3778094 q^{-37} +3175544 q^{-38} +942327 q^{-39} -1691930 q^{-40} -3373205 q^{-41} -3244729 q^{-42} -1469069 q^{-43} +962632 q^{-44} +2808633 q^{-45} +3139347 q^{-46} +1886766 q^{-47} -224497 q^{-48} -2115194 q^{-49} -2838187 q^{-50} -2127516 q^{-51} -438709 q^{-52} +1355500 q^{-53} +2354958 q^{-54} +2150543 q^{-55} +946552 q^{-56} -617661 q^{-57} -1746377 q^{-58} -1954553 q^{-59} -1238823 q^{-60} -6866 q^{-61} +1097392 q^{-62} +1583164 q^{-63} +1299853 q^{-64} +447271 q^{-65} -504567 q^{-66} -1116612 q^{-67} -1160225 q^{-68} -672613 q^{-69} +46196 q^{-70} +648950 q^{-71} +889664 q^{-72} +700205 q^{-73} +237065 q^{-74} -260227 q^{-75} -574945 q^{-76} -587886 q^{-77} -349646 q^{-78} -1773 q^{-79} +292569 q^{-80} +407982 q^{-81} +333928 q^{-82} +133949 q^{-83} -89221 q^{-84} -229710 q^{-85} -249493 q^{-86} -162516 q^{-87} -23369 q^{-88} +93644 q^{-89} +149147 q^{-90} +132781 q^{-91} +63605 q^{-92} -14057 q^{-93} -69180 q^{-94} -83861 q^{-95} -59274 q^{-96} -18707 q^{-97} +19848 q^{-98} +41277 q^{-99} +39464 q^{-100} +23395 q^{-101} +1684 q^{-102} -15056 q^{-103} -19946 q^{-104} -16215 q^{-105} -7040 q^{-106} +2517 q^{-107} +7576 q^{-108} +8655 q^{-109} +5620 q^{-110} +985 q^{-111} -1979 q^{-112} -3381 q^{-113} -2854 q^{-114} -1311 q^{-115} -33 q^{-116} +1072 q^{-117} +1289 q^{-118} +675 q^{-119} +178 q^{-120} -253 q^{-121} -344 q^{-122} -232 q^{-123} -197 q^{-124} +20 q^{-125} +141 q^{-126} +92 q^{-127} +39 q^{-128} -24 q^{-129} -21 q^{-130} +5 q^{-131} -26 q^{-132} -10 q^{-133} +14 q^{-134} +8 q^{-135} +5 q^{-136} -9 q^{-137} -2 q^{-138} +4 q^{-139} - q^{-140} }

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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