9 44

9_43

9_45

Visit 9 44's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 44's page at Knotilus! Visit 9 44's page at the original Knot Atlas! 9 44 Quick Notes

9 44 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13
Gauss code	-1, 4, -3, 1, -2, -9, 5, 3, -4, 2, 7, -8, 9, -5, 6, -7, 8, -6
Dowker-Thistlethwaite code	4 8 10 -14 2 -16 -6 -18 -12
Conway Notation	[22,21,2-]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-3] Hyperbolic Volume 7.40677 A-Polynomial See Data:9 44/A-polynomial

[edit Notes for 9 44's three dimensional invariants] 9_44 has girth 4. See arXiv:math.GT/0508590 and a forthcoming paper by the same authors.

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 9 44's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}-4t+7-4t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q^{2}-2q+3-3q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}-a^{4}+z^{4}a^{2}+3z^{2}a^{2}+3a^{2}-2z^{2}-2+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{7}+az^{7}+2a^{4}z^{6}+3a^{2}z^{6}+z^{6}+a^{5}z^{5}-2a^{3}z^{5}-3az^{5}-7a^{4}z^{4}-10a^{2}z^{4}-3z^{4}-3a^{5}z^{3}-a^{3}z^{3}+4az^{3}+2z^{3}a^{-1}+5a^{4}z^{2}+10a^{2}z^{2}+z^{2}a^{-2}+6z^{2}+a^{5}z+a^{3}z-az-za^{-1}-a^{4}-3a^{2}-a^{-2}-2$
The A2 invariant	$-q^{16}+2q^{8}+q^{6}+q^{4}-1-q^{-4}+q^{-6}+q^{-8}$
The G2 invariant	$q^{80}-q^{78}+2q^{76}-3q^{74}+q^{72}-4q^{68}+6q^{66}-5q^{64}+3q^{62}-q^{60}-4q^{58}+5q^{56}-4q^{54}+2q^{50}-4q^{48}+4q^{46}-4q^{42}+7q^{40}-5q^{38}+3q^{36}-2q^{32}+6q^{30}-4q^{28}+6q^{26}-2q^{24}+2q^{22}+4q^{20}-4q^{18}+4q^{16}-2q^{14}+q^{12}+3q^{10}-4q^{8}+2q^{6}-4q^{2}+5-6q^{-2}+2q^{-6}-6q^{-8}+5q^{-10}-3q^{-12}+q^{-14}+q^{-16}-3q^{-18}+2q^{-20}+q^{-24}+q^{-26}+q^{-32}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 9_44.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{9}+q^{5}+q^{-1}-q^{-3}+q^{-5}$
2	$q^{32}-q^{30}-2q^{28}+2q^{26}+q^{24}-2q^{22}+2q^{18}-q^{16}-q^{14}+2q^{12}-q^{8}+q^{6}+q^{4}-q^{2}-1+3q^{-2}-2q^{-6}+2q^{-8}+q^{-10}-q^{-12}$
3	$-q^{63}+q^{61}+2q^{59}-3q^{55}-3q^{53}+3q^{51}+4q^{49}-4q^{45}-3q^{43}+3q^{41}+5q^{39}-q^{37}-6q^{35}-3q^{33}+6q^{31}+4q^{29}-4q^{27}-4q^{25}+4q^{23}+5q^{21}-3q^{19}-4q^{17}+2q^{15}+3q^{13}-2q^{11}-3q^{9}+3q^{5}+3q^{3}-2q-4q^{-1}+2q^{-3}+7q^{-5}+3q^{-7}-7q^{-9}-4q^{-11}+5q^{-13}+4q^{-15}-2q^{-17}-5q^{-19}+3q^{-23}+q^{-25}-q^{-29}$
4	$q^{104}-q^{102}-2q^{100}+q^{96}+5q^{94}+q^{92}-3q^{90}-5q^{88}-6q^{86}+5q^{84}+7q^{82}+5q^{80}-10q^{76}-7q^{74}-q^{72}+9q^{70}+14q^{68}+q^{66}-11q^{64}-16q^{62}-5q^{60}+15q^{58}+17q^{56}+2q^{54}-18q^{52}-18q^{50}+5q^{48}+21q^{46}+14q^{44}-9q^{42}-20q^{40}-5q^{38}+14q^{36}+13q^{34}-4q^{32}-15q^{30}-5q^{28}+9q^{26}+8q^{24}-2q^{22}-9q^{20}-3q^{18}+7q^{16}+6q^{14}+2q^{12}-4q^{10}-7q^{8}-q^{6}+6q^{4}+13q^{2}+3-15q^{-2}-15q^{-4}-q^{-6}+22q^{-8}+20q^{-10}-9q^{-12}-25q^{-14}-15q^{-16}+15q^{-18}+26q^{-20}+5q^{-22}-14q^{-24}-18q^{-26}-2q^{-28}+13q^{-30}+9q^{-32}-7q^{-36}-5q^{-38}+q^{-40}+2q^{-42}+2q^{-44}-q^{-48}$
5	$-q^{155}+q^{153}+2q^{151}-q^{147}-3q^{145}-3q^{143}-q^{141}+5q^{139}+7q^{137}+4q^{135}-q^{133}-7q^{131}-11q^{129}-8q^{127}+5q^{125}+11q^{123}+13q^{121}+9q^{119}-4q^{117}-16q^{115}-20q^{113}-10q^{111}+6q^{109}+24q^{107}+29q^{105}+13q^{103}-16q^{101}-37q^{99}-34q^{97}-8q^{95}+32q^{93}+50q^{91}+32q^{89}-13q^{87}-54q^{85}-54q^{83}-12q^{81}+43q^{79}+67q^{77}+38q^{75}-25q^{73}-67q^{71}-53q^{69}+3q^{67}+59q^{65}+62q^{63}+13q^{61}-46q^{59}-62q^{57}-22q^{55}+32q^{53}+54q^{51}+25q^{49}-23q^{47}-45q^{45}-23q^{43}+18q^{41}+34q^{39}+17q^{37}-13q^{35}-26q^{33}-10q^{31}+12q^{29}+19q^{27}+8q^{25}-9q^{23}-16q^{21}-10q^{19}+4q^{17}+17q^{15}+17q^{13}+6q^{11}-14q^{9}-29q^{7}-21q^{5}+9q^{3}+41q+40q^{-1}+5q^{-3}-44q^{-5}-63q^{-7}-27q^{-9}+42q^{-11}+81q^{-13}+49q^{-15}-23q^{-17}-82q^{-19}-70q^{-21}+q^{-23}+69q^{-25}+78q^{-27}+19q^{-29}-44q^{-31}-66q^{-33}-36q^{-35}+18q^{-37}+47q^{-39}+36q^{-41}+q^{-43}-23q^{-45}-26q^{-47}-11q^{-49}+6q^{-51}+14q^{-53}+9q^{-55}-3q^{-59}-4q^{-61}-2q^{-63}+q^{-65}+q^{-67}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+2q^{8}+q^{6}+q^{4}-1-q^{-4}+q^{-6}+q^{-8}$
1,1	$q^{44}-2q^{42}+4q^{40}-8q^{38}+11q^{36}-12q^{34}+12q^{32}-12q^{30}+4q^{28}+2q^{26}-8q^{24}+16q^{22}-19q^{20}+22q^{18}-20q^{16}+20q^{14}-14q^{12}+10q^{10}-2q^{8}-4q^{6}+8q^{4}-14q^{2}+14-12q^{-2}+11q^{-4}-4q^{-6}+4q^{-8}+2q^{-10}-q^{-12}-2q^{-16}+q^{-20}$
2,0	$q^{42}-q^{38}-q^{36}+q^{32}-q^{30}-q^{28}+2q^{18}+3q^{16}+q^{14}+q^{12}-q^{8}-2q^{6}-q^{4}-2q^{2}+2q^{-2}+3q^{-4}+2q^{-6}+2q^{-10}-2q^{-14}-q^{-16}+q^{-20}$

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-q^{44}+q^{42}-2q^{40}+2q^{38}-2q^{36}-2q^{32}-q^{28}-q^{26}+q^{24}-q^{22}+3q^{20}+5q^{16}+5q^{12}-q^{10}+3q^{8}-q^{6}+q^{4}-2q^{2}-1-q^{-2}-2q^{-4}+q^{-6}-2q^{-8}+q^{-10}-q^{-12}+3q^{-14}+q^{-18}+q^{-22}$

G2 Invariants.

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

.

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 44"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{7}+az^{7}+2a^{4}z^{6}+3a^{2}z^{6}+z^{6}+a^{5}z^{5}-2a^{3}z^{5}-3az^{5}-7a^{4}z^{4}-10a^{2}z^{4}-3z^{4}-3a^{5}z^{3}-a^{3}z^{3}+4az^{3}+2z^{3}a^{-1}+5a^{4}z^{2}+10a^{2}z^{2}+z^{2}a^{-2}+6z^{2}+a^{5}z+a^{3}z-az-za^{-1}-a^{4}-3a^{2}-a^{-2}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(0, -1)

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 $0$ $-8$ $0$ $0$ $-8$ $0$ ${\frac {16}{3}}$ ${\frac {64}{3}}$ $-8$ $0$ $32$ $0$ $0$ $48$ $-{\frac {56}{3}}$ ${\frac {104}{3}}$ $-{\frac {16}{3}}$ $0$

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 $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 9 44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$-q^{5}+2q^{4}+q^{3}-5q^{2}+4q+4-9q^{-1}+4q^{-2}+6q^{-3}-9q^{-4}+2q^{-5}+7q^{-6}-7q^{-7}-q^{-8}+7q^{-9}-4q^{-10}-3q^{-11}+5q^{-12}-q^{-13}-2q^{-14}+q^{-15}$
3	$-q^{13}+q^{12}+q^{11}+2q^{10}-4q^{9}-4q^{8}+4q^{7}+8q^{6}-3q^{5}-13q^{4}+q^{3}+18q^{2}+q-18-5q^{-1}+20q^{-2}+6q^{-3}-18q^{-4}-8q^{-5}+17q^{-6}+7q^{-7}-13q^{-8}-9q^{-9}+11q^{-10}+8q^{-11}-5q^{-12}-10q^{-13}+3q^{-14}+8q^{-15}+3q^{-16}-8q^{-17}-6q^{-18}+5q^{-19}+8q^{-20}-2q^{-21}-8q^{-22}-q^{-23}+7q^{-24}+2q^{-25}-4q^{-26}-2q^{-27}+q^{-28}+2q^{-29}-q^{-30}$
4	$-q^{22}+q^{21}+2q^{20}-q^{18}-7q^{17}-q^{16}+9q^{15}+9q^{14}+3q^{13}-22q^{12}-17q^{11}+13q^{10}+28q^{9}+24q^{8}-33q^{7}-47q^{6}+3q^{5}+44q^{4}+53q^{3}-31q^{2}-70q-11+44q^{-1}+71q^{-2}-21q^{-3}-77q^{-4}-18q^{-5}+38q^{-6}+74q^{-7}-15q^{-8}-73q^{-9}-17q^{-10}+28q^{-11}+68q^{-12}-8q^{-13}-63q^{-14}-16q^{-15}+14q^{-16}+58q^{-17}+3q^{-18}-46q^{-19}-15q^{-20}-5q^{-21}+43q^{-22}+14q^{-23}-23q^{-24}-8q^{-25}-21q^{-26}+20q^{-27}+14q^{-28}-3q^{-29}+7q^{-30}-23q^{-31}+3q^{-33}+2q^{-34}+19q^{-35}-10q^{-36}-5q^{-37}-7q^{-38}-4q^{-39}+16q^{-40}-5q^{-43}-6q^{-44}+5q^{-45}+q^{-46}+2q^{-47}-q^{-48}-2q^{-49}+q^{-50}$
5	$q^{31}-3q^{29}-2q^{28}+q^{27}+3q^{26}+10q^{25}+5q^{24}-11q^{23}-19q^{22}-14q^{21}+6q^{20}+34q^{19}+40q^{18}-48q^{16}-68q^{15}-24q^{14}+56q^{13}+103q^{12}+59q^{11}-57q^{10}-136q^{9}-95q^{8}+44q^{7}+162q^{6}+131q^{5}-25q^{4}-175q^{3}-164q^{2}+8q+181+180q^{-1}+10q^{-2}-174q^{-3}-196q^{-4}-22q^{-5}+173q^{-6}+195q^{-7}+30q^{-8}-163q^{-9}-196q^{-10}-35q^{-11}+159q^{-12}+189q^{-13}+37q^{-14}-146q^{-15}-185q^{-16}-42q^{-17}+137q^{-18}+173q^{-19}+50q^{-20}-116q^{-21}-168q^{-22}-58q^{-23}+96q^{-24}+151q^{-25}+72q^{-26}-68q^{-27}-139q^{-28}-80q^{-29}+42q^{-30}+111q^{-31}+88q^{-32}-9q^{-33}-90q^{-34}-83q^{-35}-14q^{-36}+55q^{-37}+74q^{-38}+33q^{-39}-27q^{-40}-54q^{-41}-38q^{-42}+32q^{-44}+33q^{-45}+14q^{-46}-9q^{-47}-20q^{-48}-18q^{-49}-8q^{-50}+7q^{-51}+11q^{-52}+12q^{-53}+9q^{-54}-2q^{-55}-13q^{-56}-11q^{-57}-5q^{-58}+2q^{-59}+13q^{-60}+10q^{-61}-7q^{-63}-7q^{-64}-4q^{-65}+7q^{-67}+4q^{-68}-q^{-69}-2q^{-70}-q^{-71}-2q^{-72}+q^{-73}+2q^{-74}-q^{-75}$
6	$q^{47}-q^{46}-q^{45}-q^{42}-q^{41}+8q^{40}+3q^{39}-2q^{37}-8q^{36}-17q^{35}-16q^{34}+17q^{33}+26q^{32}+31q^{31}+25q^{30}-6q^{29}-65q^{28}-88q^{27}-25q^{26}+31q^{25}+100q^{24}+134q^{23}+82q^{22}-83q^{21}-213q^{20}-173q^{19}-66q^{18}+128q^{17}+297q^{16}+285q^{15}+9q^{14}-288q^{13}-365q^{12}-266q^{11}+47q^{10}+399q^{9}+506q^{8}+182q^{7}-259q^{6}-478q^{5}-452q^{4}-93q^{3}+397q^{2}+625q+328-179q^{-1}-490q^{-2}-539q^{-3}-199q^{-4}+349q^{-5}+645q^{-6}+390q^{-7}-120q^{-8}-460q^{-9}-549q^{-10}-240q^{-11}+310q^{-12}+624q^{-13}+399q^{-14}-96q^{-15}-428q^{-16}-530q^{-17}-248q^{-18}+279q^{-19}+587q^{-20}+395q^{-21}-71q^{-22}-384q^{-23}-500q^{-24}-261q^{-25}+221q^{-26}+524q^{-27}+395q^{-28}-12q^{-29}-301q^{-30}-454q^{-31}-296q^{-32}+115q^{-33}+420q^{-34}+391q^{-35}+84q^{-36}-168q^{-37}-372q^{-38}-330q^{-39}-28q^{-40}+264q^{-41}+347q^{-42}+175q^{-43}-q^{-44}-234q^{-45}-311q^{-46}-151q^{-47}+75q^{-48}+231q^{-49}+193q^{-50}+135q^{-51}-59q^{-52}-203q^{-53}-182q^{-54}-74q^{-55}+73q^{-56}+109q^{-57}+163q^{-58}+67q^{-59}-51q^{-60}-105q^{-61}-105q^{-62}-29q^{-63}-11q^{-64}+85q^{-65}+76q^{-66}+36q^{-67}-7q^{-68}-43q^{-69}-22q^{-70}-58q^{-71}+2q^{-72}+16q^{-73}+25q^{-74}+18q^{-75}+8q^{-76}+25q^{-77}-28q^{-78}-10q^{-79}-16q^{-80}-6q^{-81}-7q^{-82}+2q^{-83}+33q^{-84}+7q^{-86}-5q^{-87}-6q^{-88}-15q^{-89}-10q^{-90}+12q^{-91}+q^{-92}+8q^{-93}+3q^{-94}+3q^{-95}-7q^{-96}-6q^{-97}+3q^{-98}-2q^{-99}+2q^{-100}+q^{-101}+2q^{-102}-q^{-103}-2q^{-104}+q^{-105}$
7	$q^{63}-q^{62}-q^{61}-q^{60}+2q^{58}+q^{57}+2q^{56}+5q^{55}+q^{54}-5q^{53}-11q^{52}-14q^{51}-3q^{50}+q^{49}+14q^{48}+34q^{47}+36q^{46}+16q^{45}-23q^{44}-66q^{43}-74q^{42}-62q^{41}-14q^{40}+83q^{39}+152q^{38}+166q^{37}+84q^{36}-76q^{35}-217q^{34}-294q^{33}-243q^{32}-15q^{31}+257q^{30}+461q^{29}+466q^{28}+182q^{27}-225q^{26}-602q^{25}-725q^{24}-446q^{23}+93q^{22}+683q^{21}+1000q^{20}+770q^{19}+109q^{18}-684q^{17}-1216q^{16}-1089q^{15}-386q^{14}+595q^{13}+1362q^{12}+1380q^{11}+672q^{10}-460q^{9}-1421q^{8}-1593q^{7}-911q^{6}+284q^{5}+1403q^{4}+1723q^{3}+1113q^{2}-125q-1358-1784q^{-1}-1228q^{-2}+3q^{-3}+1281q^{-4}+1788q^{-5}+1300q^{-6}+89q^{-7}-1221q^{-8}-1778q^{-9}-1318q^{-10}-135q^{-11}+1171q^{-12}+1745q^{-13}+1319q^{-14}+163q^{-15}-1133q^{-16}-1720q^{-17}-1309q^{-18}-173q^{-19}+1103q^{-20}+1689q^{-21}+1291q^{-22}+187q^{-23}-1064q^{-24}-1655q^{-25}-1278q^{-26}-208q^{-27}+1010q^{-28}+1608q^{-29}+1266q^{-30}+250q^{-31}-934q^{-32}-1544q^{-33}-1252q^{-34}-313q^{-35}+818q^{-36}+1456q^{-37}+1250q^{-38}+398q^{-39}-679q^{-40}-1339q^{-41}-1222q^{-42}-503q^{-43}+484q^{-44}+1189q^{-45}+1197q^{-46}+617q^{-47}-285q^{-48}-1000q^{-49}-1117q^{-50}-717q^{-51}+45q^{-52}+762q^{-53}+1020q^{-54}+791q^{-55}+171q^{-56}-512q^{-57}-842q^{-58}-798q^{-59}-377q^{-60}+236q^{-61}+635q^{-62}+751q^{-63}+500q^{-64}+10q^{-65}-380q^{-66}-617q^{-67}-557q^{-68}-209q^{-69}+139q^{-70}+428q^{-71}+514q^{-72}+321q^{-73}+77q^{-74}-220q^{-75}-402q^{-76}-337q^{-77}-211q^{-78}+28q^{-79}+238q^{-80}+277q^{-81}+260q^{-82}+108q^{-83}-88q^{-84}-164q^{-85}-227q^{-86}-163q^{-87}-23q^{-88}+42q^{-89}+146q^{-90}+154q^{-91}+79q^{-92}+34q^{-93}-63q^{-94}-94q^{-95}-63q^{-96}-76q^{-97}-12q^{-98}+41q^{-99}+37q^{-100}+64q^{-101}+22q^{-102}+4q^{-103}+15q^{-104}-33q^{-105}-32q^{-106}-18q^{-107}-23q^{-108}+8q^{-109}+2q^{-110}+4q^{-111}+37q^{-112}+12q^{-113}+6q^{-114}-20q^{-116}-7q^{-117}-13q^{-118}-15q^{-119}+7q^{-120}+9q^{-121}+12q^{-122}+12q^{-123}-5q^{-124}+q^{-125}-3q^{-126}-10q^{-127}-3q^{-128}-2q^{-129}+4q^{-130}+6q^{-131}-q^{-132}+2q^{-134}-2q^{-135}-q^{-136}-2q^{-137}+q^{-138}+2q^{-139}-q^{-140}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 44]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[14, 8, 15, 7], X[18, 15, 1, 16], X[16, 11, 17, 12], X[12, 17, 13, 18], X[6, 14, 7, 13]]
In[3]:=	GaussCode[Knot[9, 44]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, -9, 5, 3, -4, 2, 7, -8, 9, -5, 6, -7, 8, -6]
In[4]:=	DTCode[Knot[9, 44]]
Out[4]=	DTCode[4, 8, 10, -14, 2, -16, -6, -18, -12]
In[5]:=	br = BR[Knot[9, 44]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, 1, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 44]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 44]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 44]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, {4, 5}, 1}
In[10]:=	alex = Alexander[Knot[9, 44]][t]
Out[10]=	-2 4 2 7 + t - - - 4 t + t t
In[11]:=	Conway[Knot[9, 44]][z]
Out[11]=	4 1 + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 44]}
In[13]:=	{KnotDet[Knot[9, 44]], KnotSignature[Knot[9, 44]]}
Out[13]=	{17, 0}
In[14]:=	Jones[Knot[9, 44]][q]
Out[14]=	-5 2 2 3 3 2 3 - q + -- - -- + -- - - - 2 q + q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 44]}
In[16]:=	A2Invariant[Knot[9, 44]][q]
Out[16]=	-16 2 -6 -4 4 6 8 -1 - q + -- + q + q - q + q + q 8 q
In[17]:=	HOMFLYPT[Knot[9, 44]][a, z]
Out[17]=	-2 2 4 2 2 2 4 2 2 4 -2 + a + 3 a - a - 2 z + 3 a z - a z + a z
In[18]:=	Kauffman[Knot[9, 44]][a, z]
Out[18]=	2 -2 2 4 z 3 5 2 z 2 2 -2 - a - 3 a - a - - - a z + a z + a z + 6 z + -- + 10 a z + a 2 a 3 4 2 2 z 3 3 3 5 3 4 2 4 5 a z + ---- + 4 a z - a z - 3 a z - 3 z - 10 a z - a 4 4 5 3 5 5 5 6 2 6 4 6 7 7 a z - 3 a z - 2 a z + a z + z + 3 a z + 2 a z + a z + 3 7 a z
In[19]:=	{Vassiliev[2][Knot[9, 44]], Vassiliev[3][Knot[9, 44]]}
Out[19]=	{0, -1}
In[20]:=	Kh[Knot[9, 44]][q, t]
Out[20]=	2 1 1 1 1 1 2 1 - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 1 2 3 5 2 ---- + --- + q t + q t + q t 3 q t q t
In[21]:=	ColouredJones[Knot[9, 44], 2][q]
Out[21]=	-15 2 -13 5 3 4 7 -8 7 7 2 4 + q - --- - q + --- - --- - --- + -- - q - -- + -- + -- - 14 12 11 10 9 7 6 5 q q q q q q q q 9 6 4 9 2 3 4 5 -- + -- + -- - - + 4 q - 5 q + q + 2 q - q 4 3 2 q q q q

See/edit the Rolfsen_Splice_Template. Back to the top.

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