9 43

9_42

9_44

Visit 9 43's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 43's page at Knotilus! Visit 9 43's page at the original Knot Atlas! 9 43 Quick Notes

9 43 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X15,1,16,18 X11,17,12,16 X17,13,18,12 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	[211,3,2-]

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 2 3-genus 3 Bridge index 3 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [1][-10] Hyperbolic Volume 5.90409 A-Polynomial See Data:9 43/A-polynomial

[edit Notes for 9 43's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for 9 43's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+3t^{2}-2t+1-2t^{-1}+3t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 13, 4 }
Jones polynomial	$-q^{7}+2q^{6}-2q^{5}+2q^{4}-2q^{3}+2q^{2}-q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}+z^{4}a^{-2}-5z^{4}a^{-4}+z^{4}a^{-6}+4z^{2}a^{-2}-7z^{2}a^{-4}+4z^{2}a^{-6}+3a^{-2}-4a^{-4}+3a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-3}+z^{7}a^{-5}+z^{6}a^{-2}+3z^{6}a^{-4}+2z^{6}a^{-6}-4z^{5}a^{-3}-3z^{5}a^{-5}+z^{5}a^{-7}-5z^{4}a^{-2}-13z^{4}a^{-4}-8z^{4}a^{-6}+3z^{3}a^{-3}+z^{3}a^{-5}-2z^{3}a^{-7}+7z^{2}a^{-2}+14z^{2}a^{-4}+9z^{2}a^{-6}+2z^{2}a^{-8}+za^{-7}+za^{-9}-3a^{-2}-4a^{-4}-3a^{-6}-a^{-8}$
The A2 invariant	$1+q^{-2}+q^{-4}+q^{-6}-2q^{-12}+q^{-18}+q^{-20}-q^{-26}$
The G2 invariant	$q^{-2}+2q^{-6}-q^{-8}+q^{-10}+q^{-12}-q^{-14}+4q^{-16}-q^{-18}+2q^{-20}+q^{-22}-q^{-24}+3q^{-26}-q^{-30}+2q^{-32}-q^{-34}+2q^{-38}-3q^{-40}+2q^{-42}-2q^{-44}-q^{-48}-3q^{-50}+q^{-52}-3q^{-54}+q^{-56}-2q^{-58}-q^{-64}+q^{-66}-q^{-68}+2q^{-72}+3q^{-78}-2q^{-80}+4q^{-82}+2q^{-88}-2q^{-90}+2q^{-92}-q^{-100}-q^{-102}+q^{-104}-q^{-106}-q^{-108}-q^{-112}-q^{-116}+q^{-120}$

Further Quantum Invariants

Further quantum knot invariants for 9_43.

A1 Invariants.

Weight	Invariant
1	$q+q^{-3}+q^{-13}-q^{-15}$
2	$q^{6}-q^{2}+1+q^{-2}-q^{-4}+q^{-8}+q^{-10}+q^{-14}+q^{-16}-q^{-18}+q^{-22}-q^{-24}-q^{-26}-q^{-32}+q^{-36}$
3	$q^{15}-q^{11}-q^{9}+q^{7}+2q^{5}-2q-q^{-1}+q^{-3}+2q^{-5}+q^{-7}-q^{-11}+q^{-13}+3q^{-15}+q^{-17}-2q^{-19}-2q^{-21}+2q^{-23}+2q^{-25}-q^{-27}-2q^{-29}+q^{-31}+2q^{-33}-q^{-35}-2q^{-37}+q^{-39}+q^{-41}-2q^{-43}-2q^{-45}+q^{-49}+q^{-51}-2q^{-55}-q^{-57}+3q^{-59}+3q^{-61}-2q^{-63}-4q^{-65}+2q^{-67}+3q^{-69}-2q^{-73}-q^{-75}+q^{-77}$
4	$q^{28}-q^{24}-q^{22}-q^{20}+2q^{18}+2q^{16}+q^{14}-q^{12}-4q^{10}-q^{8}+q^{6}+3q^{4}+3q^{2}-q^{-2}-3q^{-4}-2q^{-6}+2q^{-8}+4q^{-10}+4q^{-12}-5q^{-16}-4q^{-18}+q^{-20}+6q^{-22}+6q^{-24}-2q^{-26}-5q^{-28}-4q^{-30}+2q^{-32}+8q^{-34}+3q^{-36}-3q^{-38}-6q^{-40}-3q^{-42}+5q^{-44}+4q^{-46}-q^{-48}-5q^{-50}-4q^{-52}+3q^{-54}+3q^{-56}-q^{-58}-3q^{-60}-2q^{-62}+3q^{-64}+2q^{-66}-q^{-68}-3q^{-70}-2q^{-72}+3q^{-74}+3q^{-76}+q^{-78}-q^{-80}-4q^{-82}-3q^{-84}+3q^{-86}+7q^{-88}+5q^{-90}-4q^{-92}-10q^{-94}-3q^{-96}+6q^{-98}+10q^{-100}+2q^{-102}-10q^{-104}-6q^{-106}+6q^{-110}+4q^{-112}-3q^{-114}-2q^{-116}-q^{-118}+2q^{-120}+q^{-122}-q^{-124}$
5	$q^{45}-q^{41}-q^{39}-q^{37}+2q^{33}+3q^{31}+q^{29}-q^{27}-3q^{25}-4q^{23}-2q^{21}+2q^{19}+5q^{17}+4q^{15}+2q^{13}-q^{11}-4q^{9}-5q^{7}-3q^{5}+4q+7q^{-1}+6q^{-3}+q^{-5}-6q^{-7}-9q^{-9}-7q^{-11}+9q^{-15}+13q^{-17}+8q^{-19}-3q^{-21}-11q^{-23}-12q^{-25}-4q^{-27}+8q^{-29}+16q^{-31}+12q^{-33}-12q^{-37}-16q^{-39}-8q^{-41}+6q^{-43}+16q^{-45}+13q^{-47}-13q^{-51}-16q^{-53}-5q^{-55}+9q^{-57}+16q^{-59}+9q^{-61}-6q^{-63}-15q^{-65}-12q^{-67}+2q^{-69}+12q^{-71}+11q^{-73}-11q^{-77}-11q^{-79}-q^{-81}+8q^{-83}+8q^{-85}-7q^{-89}-6q^{-91}+3q^{-93}+7q^{-95}+4q^{-97}-4q^{-99}-7q^{-101}-2q^{-103}+5q^{-105}+8q^{-107}+4q^{-109}-3q^{-111}-7q^{-113}-7q^{-115}-2q^{-117}+6q^{-119}+11q^{-121}+9q^{-123}+q^{-125}-10q^{-127}-17q^{-129}-10q^{-131}+6q^{-133}+18q^{-135}+17q^{-137}+2q^{-139}-17q^{-141}-24q^{-143}-11q^{-145}+12q^{-147}+22q^{-149}+16q^{-151}-2q^{-153}-16q^{-155}-15q^{-157}-2q^{-159}+9q^{-161}+9q^{-163}+3q^{-165}-3q^{-167}-4q^{-169}-2q^{-171}+q^{-173}+q^{-175}$

A2 Invariants.

Weight	Invariant
1,0	$1+q^{-2}+q^{-4}+q^{-6}-2q^{-12}+q^{-18}+q^{-20}-q^{-26}$
1,1	$q^{4}+4-2q^{-2}+6q^{-4}-4q^{-6}+4q^{-8}-4q^{-10}-q^{-12}+2q^{-14}-4q^{-16}+6q^{-18}-4q^{-20}+10q^{-22}-4q^{-24}+8q^{-26}-4q^{-28}+2q^{-30}-4q^{-32}-4q^{-34}-6q^{-38}+4q^{-40}-2q^{-42}+4q^{-44}+2q^{-48}-q^{-52}-2q^{-54}+q^{-60}$
2,0	$q^{4}+q^{2}+1+q^{-4}+q^{-6}-q^{-10}+q^{-20}+q^{-22}+q^{-24}+q^{-26}+2q^{-28}+2q^{-30}+q^{-32}-3q^{-36}-3q^{-38}-3q^{-40}-2q^{-42}-q^{-44}+2q^{-48}+3q^{-50}+2q^{-52}-q^{-58}-q^{-60}-q^{-62}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$1+2q^{-4}+2q^{-6}+2q^{-8}+2q^{-10}+q^{-12}-q^{-16}-2q^{-18}-2q^{-20}-q^{-22}-q^{-24}+q^{-28}+q^{-30}+2q^{-32}+q^{-34}+q^{-36}-q^{-40}-q^{-42}-q^{-44}-q^{-46}+q^{-50}$
1,0,0	$q^{-1}+q^{-3}+2q^{-5}+q^{-7}+2q^{-9}-q^{-13}-2q^{-15}-2q^{-17}-q^{-19}+2q^{-23}+q^{-25}+2q^{-27}-q^{-33}-q^{-35}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-4}+2q^{-6}+3q^{-8}+4q^{-10}+4q^{-12}+3q^{-14}+q^{-16}-2q^{-20}-4q^{-22}-4q^{-24}-2q^{-26}-q^{-28}+2q^{-34}+q^{-36}-q^{-38}+q^{-42}+2q^{-46}+3q^{-48}+q^{-50}-q^{-56}-3q^{-58}-2q^{-60}-q^{-66}+q^{-68}+q^{-70}$
1,0,0,0	$q^{-2}+q^{-4}+2q^{-6}+2q^{-8}+2q^{-10}+2q^{-12}-q^{-16}-3q^{-18}-2q^{-20}-3q^{-22}-q^{-24}+2q^{-28}+2q^{-30}+2q^{-32}+2q^{-34}-q^{-40}-q^{-42}-q^{-44}$

B2 Invariants.

Weight	Invariant
0,1	$1+2q^{-4}+2q^{-8}+q^{-12}-q^{-16}-2q^{-20}+q^{-22}-3q^{-24}+2q^{-26}-q^{-28}+q^{-30}+q^{-34}+q^{-36}+q^{-40}-q^{-42}+q^{-44}-q^{-46}-q^{-50}$
1,0	$q^{2}+2q^{-6}+q^{-8}+2q^{-14}+q^{-16}-q^{-20}+q^{-24}-q^{-28}-q^{-30}-q^{-38}+q^{-42}-q^{-46}+q^{-50}+q^{-52}-q^{-56}+q^{-58}+q^{-60}-q^{-64}-q^{-72}-q^{-74}+q^{-80}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-2}+2q^{-6}+q^{-8}+4q^{-10}+2q^{-12}+3q^{-14}+2q^{-16}+2q^{-18}-2q^{-22}-2q^{-24}-4q^{-26}-2q^{-28}-4q^{-30}-q^{-32}-3q^{-34}+2q^{-36}+3q^{-40}+2q^{-42}+3q^{-44}+2q^{-46}+2q^{-48}+q^{-50}-q^{-52}-2q^{-56}-q^{-58}-2q^{-60}-q^{-64}+q^{-70}$

G2 Invariants.

Weight	Invariant
1,0	$q^{-2}+2q^{-6}-q^{-8}+q^{-10}+q^{-12}-q^{-14}+4q^{-16}-q^{-18}+2q^{-20}+q^{-22}-q^{-24}+3q^{-26}-q^{-30}+2q^{-32}-q^{-34}+2q^{-38}-3q^{-40}+2q^{-42}-2q^{-44}-q^{-48}-3q^{-50}+q^{-52}-3q^{-54}+q^{-56}-2q^{-58}-q^{-64}+q^{-66}-q^{-68}+2q^{-72}+3q^{-78}-2q^{-80}+4q^{-82}+2q^{-88}-2q^{-90}+2q^{-92}-q^{-100}-q^{-102}+q^{-104}-q^{-106}-q^{-108}-q^{-112}-q^{-116}+q^{-120}$

.

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 43"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+3t^{2}-2t+1-2t^{-1}+3t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 13, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-3}+z^{7}a^{-5}+z^{6}a^{-2}+3z^{6}a^{-4}+2z^{6}a^{-6}-4z^{5}a^{-3}-3z^{5}a^{-5}+z^{5}a^{-7}-5z^{4}a^{-2}-13z^{4}a^{-4}-8z^{4}a^{-6}+3z^{3}a^{-3}+z^{3}a^{-5}-2z^{3}a^{-7}+7z^{2}a^{-2}+14z^{2}a^{-4}+9z^{2}a^{-6}+2z^{2}a^{-8}+za^{-7}+za^{-9}-3a^{-2}-4a^{-4}-3a^{-6}-a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n12, ...}

Vassiliev invariants

V2 and V3:

(1, 2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $4$ $16$ $8$ ${\frac {254}{3}}$ ${\frac {82}{3}}$ $64$ ${\frac {1024}{3}}$ ${\frac {256}{3}}$ $80$ ${\frac {32}{3}}$ $128$ ${\frac {1016}{3}}$ ${\frac {328}{3}}$ ${\frac {37231}{30}}$ $-{\frac {154}{5}}$ ${\frac {31502}{45}}$ ${\frac {593}{18}}$ ${\frac {2671}{30}}$

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 $j$, 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=$4 is the signature of 9 43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{17}-q^{16}-q^{15}+2q^{14}-q^{13}-2q^{12}+2q^{11}+q^{10}-3q^{9}+q^{8}+3q^{7}-3q^{6}+4q^{4}-3q^{3}-q^{2}+3q-1-q^{-1}+q^{-2}$
3	$q^{37}-2q^{36}-q^{35}+2q^{34}+4q^{33}-3q^{32}-7q^{31}+4q^{30}+9q^{29}-3q^{28}-11q^{27}+3q^{26}+11q^{25}-2q^{24}-11q^{23}+2q^{22}+9q^{21}-2q^{20}-8q^{19}+2q^{18}+6q^{17}-q^{16}-5q^{15}+q^{14}+3q^{13}-2q^{11}+q^{10}-q^{9}+q^{7}+3q^{6}-3q^{5}-2q^{4}+2q^{3}+4q^{2}-2q-3+3q^{-2}-q^{-4}-q^{-5}+q^{-6}$
4	$-q^{60}+2q^{59}+q^{58}-3q^{57}-q^{56}-2q^{55}+9q^{54}+3q^{53}-9q^{52}-7q^{51}-6q^{50}+21q^{49}+11q^{48}-13q^{47}-16q^{46}-13q^{45}+27q^{44}+20q^{43}-11q^{42}-20q^{41}-19q^{40}+26q^{39}+23q^{38}-9q^{37}-18q^{36}-19q^{35}+21q^{34}+22q^{33}-7q^{32}-15q^{31}-18q^{30}+16q^{29}+21q^{28}-5q^{27}-11q^{26}-18q^{25}+9q^{24}+20q^{23}-q^{22}-6q^{21}-17q^{20}+q^{19}+17q^{18}+2q^{17}-12q^{15}-5q^{14}+11q^{13}+q^{12}+3q^{11}-4q^{10}-5q^{9}+6q^{8}-4q^{7}+2q^{6}+q^{5}-q^{4}+6q^{3}-6q^{2}-2q+q^{-1}+7q^{-2}-3q^{-3}-2q^{-4}-2q^{-5}-q^{-6}+4q^{-7}-q^{-10}-q^{-11}+q^{-12}$
5	$q^{85}-3q^{83}-2q^{82}+q^{81}+6q^{80}+7q^{79}-14q^{77}-15q^{76}+20q^{74}+25q^{73}+6q^{72}-24q^{71}-38q^{70}-13q^{69}+27q^{68}+44q^{67}+21q^{66}-23q^{65}-50q^{64}-29q^{63}+20q^{62}+51q^{61}+32q^{60}-15q^{59}-48q^{58}-34q^{57}+12q^{56}+46q^{55}+32q^{54}-11q^{53}-41q^{52}-30q^{51}+9q^{50}+39q^{49}+27q^{48}-8q^{47}-33q^{46}-27q^{45}+5q^{44}+30q^{43}+26q^{42}-q^{41}-25q^{40}-27q^{39}-4q^{38}+20q^{37}+26q^{36}+10q^{35}-14q^{34}-26q^{33}-14q^{32}+6q^{31}+23q^{30}+19q^{29}+q^{28}-19q^{27}-21q^{26}-8q^{25}+12q^{24}+22q^{23}+14q^{22}-6q^{21}-18q^{20}-18q^{19}-2q^{18}+14q^{17}+18q^{16}+6q^{15}-6q^{14}-14q^{13}-10q^{12}+2q^{11}+10q^{10}+7q^{9}+2q^{8}-3q^{7}-5q^{6}-2q^{5}+q^{4}+2+3q^{-1}+2q^{-2}-3q^{-3}-4q^{-4}-3q^{-5}+4q^{-7}+5q^{-8}-2q^{-10}-2q^{-11}-3q^{-12}+3q^{-14}+q^{-15}-q^{-18}-q^{-19}+q^{-20}$
6	$q^{122}-2q^{121}-q^{120}+2q^{119}+q^{118}+2q^{117}-q^{116}+2q^{115}-9q^{114}-8q^{113}+6q^{112}+9q^{111}+14q^{110}+6q^{109}+q^{108}-35q^{107}-34q^{106}+q^{105}+25q^{104}+49q^{103}+38q^{102}+16q^{101}-70q^{100}-85q^{99}-31q^{98}+24q^{97}+86q^{96}+87q^{95}+55q^{94}-81q^{93}-124q^{92}-74q^{91}-3q^{90}+92q^{89}+117q^{88}+96q^{87}-64q^{86}-128q^{85}-95q^{84}-28q^{83}+77q^{82}+115q^{81}+110q^{80}-50q^{79}-116q^{78}-91q^{77}-32q^{76}+68q^{75}+103q^{74}+103q^{73}-49q^{72}-106q^{71}-80q^{70}-26q^{69}+65q^{68}+93q^{67}+93q^{66}-48q^{65}-95q^{64}-71q^{63}-27q^{62}+55q^{61}+81q^{60}+89q^{59}-36q^{58}-76q^{57}-64q^{56}-36q^{55}+35q^{54}+64q^{53}+86q^{52}-15q^{51}-48q^{50}-54q^{49}-47q^{48}+8q^{47}+40q^{46}+79q^{45}+8q^{44}-15q^{43}-36q^{42}-51q^{41}-18q^{40}+8q^{39}+59q^{38}+21q^{37}+17q^{36}-7q^{35}-38q^{34}-30q^{33}-23q^{32}+24q^{31}+14q^{30}+32q^{29}+22q^{28}-8q^{27}-17q^{26}-35q^{25}-10q^{24}-11q^{23}+20q^{22}+30q^{21}+17q^{20}+12q^{19}-20q^{18}-19q^{17}-27q^{16}-6q^{15}+12q^{14}+16q^{13}+28q^{12}+2q^{11}-4q^{10}-17q^{9}-15q^{8}-6q^{7}-q^{6}+18q^{5}+4q^{4}+6q^{3}-4q-4-6q^{-1}+7q^{-2}-7q^{-3}+q^{-5}+2q^{-6}+3q^{-7}+q^{-8}+9q^{-9}-7q^{-10}-4q^{-11}-4q^{-12}-2q^{-13}+2q^{-15}+9q^{-16}-q^{-17}-2q^{-19}-2q^{-20}-3q^{-21}-q^{-22}+4q^{-23}+q^{-25}-q^{-28}-q^{-29}+q^{-30}$
7	$-q^{161}+2q^{160}+q^{159}-2q^{158}-2q^{157}-2q^{156}+4q^{155}+2q^{154}+q^{153}+5q^{152}-q^{151}-11q^{150}-13q^{149}-9q^{148}+13q^{147}+24q^{146}+21q^{145}+22q^{144}-12q^{143}-45q^{142}-57q^{141}-46q^{140}+16q^{139}+74q^{138}+96q^{137}+85q^{136}-3q^{135}-97q^{134}-142q^{133}-145q^{132}-30q^{131}+111q^{130}+194q^{129}+206q^{128}+68q^{127}-101q^{126}-220q^{125}-264q^{124}-129q^{123}+77q^{122}+236q^{121}+307q^{120}+172q^{119}-43q^{118}-221q^{117}-325q^{116}-212q^{115}+6q^{114}+204q^{113}+329q^{112}+228q^{111}+18q^{110}-181q^{109}-318q^{108}-232q^{107}-31q^{106}+164q^{105}+305q^{104}+226q^{103}+32q^{102}-156q^{101}-294q^{100}-215q^{99}-27q^{98}+154q^{97}+283q^{96}+206q^{95}+22q^{94}-153q^{93}-277q^{92}-198q^{91}-17q^{90}+150q^{89}+264q^{88}+194q^{87}+20q^{86}-141q^{85}-255q^{84}-189q^{83}-25q^{82}+127q^{81}+238q^{80}+187q^{79}+37q^{78}-109q^{77}-220q^{76}-184q^{75}-52q^{74}+86q^{73}+199q^{72}+182q^{71}+68q^{70}-57q^{69}-174q^{68}-179q^{67}-87q^{66}+29q^{65}+146q^{64}+168q^{63}+105q^{62}+5q^{61}-112q^{60}-158q^{59}-117q^{58}-34q^{57}+71q^{56}+134q^{55}+125q^{54}+67q^{53}-33q^{52}-108q^{51}-118q^{50}-85q^{49}-10q^{48}+66q^{47}+105q^{46}+99q^{45}+43q^{44}-33q^{43}-74q^{42}-88q^{41}-67q^{40}-12q^{39}+40q^{38}+75q^{37}+73q^{36}+32q^{35}-4q^{34}-36q^{33}-62q^{32}-52q^{31}-25q^{30}+7q^{29}+40q^{28}+39q^{27}+40q^{26}+27q^{25}-9q^{24}-26q^{23}-35q^{22}-39q^{21}-17q^{20}-3q^{19}+21q^{18}+41q^{17}+27q^{16}+21q^{15}+6q^{14}-25q^{13}-29q^{12}-33q^{11}-19q^{10}+9q^{9}+13q^{8}+25q^{7}+31q^{6}+8q^{5}-q^{4}-18q^{3}-23q^{2}-8q-10+15q^{-2}+8q^{-3}+10q^{-4}+3q^{-5}-6q^{-6}+3q^{-7}-4q^{-8}-5q^{-9}+q^{-10}-6q^{-11}-q^{-12}-4q^{-14}+6q^{-15}+5q^{-16}+4q^{-17}+7q^{-18}-4q^{-19}-4q^{-20}-3q^{-21}-7q^{-22}-2q^{-23}+3q^{-25}+8q^{-26}+q^{-27}+q^{-29}-3q^{-30}-2q^{-31}-3q^{-32}-q^{-33}+3q^{-34}+q^{-35}+q^{-37}-q^{-40}-q^{-41}+q^{-42}$

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, 43]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[14, 8, 15, 7], X[15, 1, 16, 18], X[11, 17, 12, 16], X[17, 13, 18, 12], X[6, 14, 7, 13]]
In[3]:=	GaussCode[Knot[9, 43]]
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, 43]]
Out[4]=	DTCode[4, 8, 10, 14, 2, -16, 6, -18, -12]
In[5]:=	br = BR[Knot[9, 43]]
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, 43]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 43]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 43]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, {4, 5}, 1}
In[10]:=	alex = Alexander[Knot[9, 43]][t]
Out[10]=	-3 3 2 2 3 1 - t + -- - - - 2 t + 3 t - t 2 t t
In[11]:=	Conway[Knot[9, 43]][z]
Out[11]=	2 4 6 1 + z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 43]}
In[13]:=	{KnotDet[Knot[9, 43]], KnotSignature[Knot[9, 43]]}
Out[13]=	{13, 4}
In[14]:=	Jones[Knot[9, 43]][q]
Out[14]=	2 3 4 5 6 7 1 - q + 2 q - 2 q + 2 q - 2 q + 2 q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 43], Knot[11, NonAlternating, 12]}
In[16]:=	A2Invariant[Knot[9, 43]][q]
Out[16]=	2 4 6 12 18 20 26 1 + q + q + q - 2 q + q + q - q
In[17]:=	HOMFLYPT[Knot[9, 43]][a, z]
Out[17]=	2 2 2 4 4 4 6 -8 3 4 3 4 z 7 z 4 z z 5 z z z -a + -- - -- + -- + ---- - ---- + ---- + -- - ---- + -- - -- 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a
In[18]:=	Kauffman[Knot[9, 43]][a, z]
Out[18]=	2 2 2 2 3 -8 3 4 3 z z 2 z 9 z 14 z 7 z 2 z -a - -- - -- - -- + -- + -- + ---- + ---- + ----- + ---- - ---- + 6 4 2 9 7 8 6 4 2 7 a a a a a a a a a a 3 3 4 4 4 5 5 5 6 6 z 3 z 8 z 13 z 5 z z 3 z 4 z 2 z 3 z -- + ---- - ---- - ----- - ---- + -- - ---- - ---- + ---- + ---- + 5 3 6 4 2 7 5 3 6 4 a a a a a a a a a a 6 7 7 z z z -- + -- + -- 2 5 3 a a a
In[19]:=	{Vassiliev[2][Knot[9, 43]], Vassiliev[3][Knot[9, 43]]}
Out[19]=	{1, 2}
In[20]:=	Kh[Knot[9, 43]][q, t]
Out[20]=	3 3 5 1 q 5 7 7 2 9 2 9 3 11 3 2 q + q + ---- + -- + q t + q t + q t + q t + q t + q t + 2 t q t 11 4 13 4 15 5 q t + q t + q t
In[21]:=	ColouredJones[Knot[9, 43], 2][q]
Out[21]=	-2 1 2 3 4 6 7 8 9 10 -1 + q - - + 3 q - q - 3 q + 4 q - 3 q + 3 q + q - 3 q + q + q 11 12 13 14 15 16 17 2 q - 2 q - q + 2 q - q - q + q

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9_42 9_44