9 43

9_42

9_44

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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-]

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]=

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

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)

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

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 43]]
Out[2]=	9
In[3]:=	PD[Knot[9, 43]]
Out[3]=	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[4]:=	GaussCode[Knot[9, 43]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, -9, 5, -3, 4, -2, -7, 8, 9, -5, -6, 7, -8, 6]
In[5]:=	BR[Knot[9, 43]]
Out[5]=	BR[4, {1, 1, 1, 2, 1, 1, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 43]][t]
Out[6]=	-3 3 2 2 3 1 - t + -- - - - 2 t + 3 t - t 2 t t
In[7]:=	Conway[Knot[9, 43]][z]
Out[7]=	2 4 6 1 + z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 43]}
In[9]:=	{KnotDet[Knot[9, 43]], KnotSignature[Knot[9, 43]]}
Out[9]=	{13, 4}
In[10]:=	J=Jones[Knot[9, 43]][q]
Out[10]=	2 3 4 5 6 7 1 - q + 2 q - 2 q + 2 q - 2 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 43], Knot[11, NonAlternating, 12]}
In[12]:=	A2Invariant[Knot[9, 43]][q]
Out[12]=	2 4 6 12 18 20 26 1 + q + q + q - 2 q + q + q - q
In[13]:=	Kauffman[Knot[9, 43]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[9, 43]], Vassiliev[3][Knot[9, 43]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[9, 43]][q, t]
Out[15]=	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