© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a8 K11a10

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X16,7,17,8 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,16,1,15 X6,17,7,18 X12,20,13,19 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 16 2 18 20 22 6 12 14

Alexander Polynomial:

t-4 - 5t-3 + 10t-2 - 11t-1 + 11 - 11t + 10t2 - 5t3 + t4

Conway Polynomial:

1 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{65, 4}

Jones Polynomial:

q-2 - 2q-1 + 4 - 6q + 8q2 - 9q3 + 10q4 - 9q5 + 7q6 - 5q7 + 3q8 - q9

Other knots (up to mirrors) with the same Jones Polynomial:

{K11a140, ...}

A2 (sl(3)) Invariant:

q-6 + q-4 + 1 - q2 - q6 - q8 + 2q10 - q12 + 3q14 - q22 + q24 - q26

HOMFLY-PT Polynomial:

- 2a-6 - 4a-6z2 - 4a-6z4 - a-6z6 + 6a-4 + 14a-4z2 + 13a-4z4 + 6a-4z6 + a-4z8 - 6a-2 - 14a-2z2 - 10a-2z4 - 2a-2z6 + 3 + 4z2 + z4

Kauffman Polynomial:

a-11z3 - a-10z2 + 3a-10z4 + a-9z - 4a-9z3 + 5a-9z5 + a-8z2 - 7a-8z4 + 6a-8z6 + a-7z - a-7z3 - 9a-7z5 + 6a-7z7 + 2a-6 - 7a-6z2 + 12a-6z4 - 16a-6z6 + 6a-6z8 + a-5z - a-5z3 + 5a-5z5 - 11a-5z7 + 4a-5z9 + 6a-4 - 29a-4z2 + 54a-4z4 - 36a-4z6 + 5a-4z8 + a-4z10 + 3a-3z - 17a-3z3 + 38a-3z5 - 28a-3z7 + 6a-3z9 + 6a-2 - 30a-2z2 + 44a-2z4 - 20a-2z6 + a-2z10 + 2a-1z - 12a-1z3 + 19a-1z5 - 11a-1z7 + 2a-1z9 + 3 - 10z2 + 12z4 - 6z6 + z8

V2 and V3, the type 2 and 3 Vassiliev invariants:

{0, 2}

Khovanov Homology: (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 119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                       1 j = 17                     2   j = 15                   3 1   j = 13                 4 2     j = 11               5 3       j = 9             5 4         j = 7           4 5           j = 5         4 5             j = 3       3 5               j = 1     1 3                 j = -1   1 3                   j = -3   1                     j = -5 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 9]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 9]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 7, 17, 8], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[6, 17, 7, 18], X[12, 20, 13, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 9]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -2, 5, -3, 6, -10, 7, -11, 8, -4, 9, -6, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 9]]
Out[5]=	DTCode[4, 8, 10, 16, 2, 18, 20, 22, 6, 12, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 9]][t]
Out[6]=	-4 5 10 11 2 3 4 11 + t - -- + -- - -- - 11 t + 10 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 9]][z]
Out[7]=	6 8 1 + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 9]}
In[9]:=	{KnotDet[Knot[11, Alternating, 9]], KnotSignature[Knot[11, Alternating, 9]]}
Out[9]=	{65, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 9]][q]
Out[10]=	-2 2 2 3 4 5 6 7 8 9 4 + q - - - 6 q + 8 q - 9 q + 10 q - 9 q + 7 q - 5 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 9], Knot[11, Alternating, 140]}
In[12]:=	A2Invariant[Knot[11, Alternating, 9]][q]
Out[12]=	-6 -4 2 6 8 10 12 14 22 24 26 1 + q + q - q - q - q + 2 q - q + 3 q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 9]][a, z]
Out[13]=	2 2 2 4 4 4 2 6 6 2 4 z 14 z 14 z 4 4 z 13 z 10 z 3 - -- + -- - -- + 4 z - ---- + ----- - ----- + z - ---- + ----- - ----- - 6 4 2 6 4 2 6 4 2 a a a a a a a a a 6 6 6 8 z 6 z 2 z z > -- + ---- - ---- + -- 6 4 2 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 9]][a, z]
Out[14]=	2 2 2 2 2 6 6 z z z 3 z 2 z 2 z z 7 z 29 z 3 + -- + -- + -- + -- + -- + -- + --- + --- - 10 z - --- + -- - ---- - ----- - 6 4 2 9 7 5 3 a 10 8 6 4 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 30 z z 4 z z z 17 z 12 z 4 3 z 7 z > ----- + --- - ---- - -- - -- - ----- - ----- + 12 z + ---- - ---- + 2 11 9 7 5 3 a 10 8 a a a a a a a a 4 4 4 5 5 5 5 5 6 12 z 54 z 44 z 5 z 9 z 5 z 38 z 19 z 6 6 z > ----- + ----- + ----- + ---- - ---- + ---- + ----- + ----- - 6 z + ---- - 6 4 2 9 7 5 3 a 8 a a a a a a a a 6 6 6 7 7 7 7 8 8 16 z 36 z 20 z 6 z 11 z 28 z 11 z 8 6 z 5 z > ----- - ----- - ----- + ---- - ----- - ----- - ----- + z + ---- + ---- + 6 4 2 7 5 3 a 6 4 a a a a a a a a 9 9 9 10 10 4 z 6 z 2 z z z > ---- + ---- + ---- + --- + --- 5 3 a 4 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 9]], Vassiliev[3][Knot[11, Alternating, 9]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 9]][q, t]
Out[16]=	3 3 5 1 1 1 3 q 3 q 3 q 5 7 5 q + 4 q + ----- + ----- + ---- + ---- + -- + --- + ---- + 5 q t + 4 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 13 5 > 5 q t + 5 q t + 4 q t + 5 q t + 3 q t + 4 q t + 2 q t + 15 5 15 6 17 6 19 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a9

K11a8 K11a10