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

K11a162 K11a164

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X18,5,19,6 X14,8,15,7 X16,10,17,9 X2,11,3,12 X20,14,21,13 X8,16,9,15 X22,17,1,18 X12,20,13,19 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 14 16 2 20 8 22 12 6

Alexander Polynomial:

- t-4 + 6t-3 - 15t-2 + 24t-1 - 27 + 24t - 15t2 + 6t3 - t4

Conway Polynomial:

1 + 2z2 + z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a66, ...}

Determinant and Signature:

{119, 2}

Jones Polynomial:

- q-4 + 3q-3 - 7q-2 + 12q-1 - 15 + 19q - 19q2 + 17q3 - 13q4 + 8q5 - 4q6 + q7

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

{...}

A2 (sl(3)) Invariant:

- q-12 - 2q-6 + 3q-4 - q-2 + 3 + 3q2 - 2q4 + 4q6 - 4q8 + 2q10 - q12 - 2q14 + 2q16 - 2q18 + q20

HOMFLY-PT Polynomial:

2a-4z2 + 3a-4z4 + a-4z6 - 2a-2 - 7a-2z2 - 9a-2z4 - 5a-2z6 - a-2z8 + 5 + 10z2 + 8z4 + 2z6 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 2a-7z3 + 4a-7z5 + a-6z2 - 7a-6z4 + 8a-6z6 + 5a-5z3 - 14a-5z5 + 11a-5z7 - 2a-4z2 + 8a-4z4 - 17a-4z6 + 11a-4z8 - 2a-3z + 12a-3z3 - 13a-3z5 - 5a-3z7 + 7a-3z9 + 2a-2 - 14a-2z2 + 40a-2z4 - 44a-2z6 + 12a-2z8 + 2a-2z10 - 4a-1z + 6a-1z3 + 13a-1z5 - 28a-1z7 + 11a-1z9 + 5 - 18z2 + 37z4 - 30z6 + 4z8 + 2z10 - 4az + 6az3 + 4az5 - 11az7 + 4az9 + 2a2 - 7a2z2 + 13a2z4 - 11a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

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

{2, 0}

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=2 is the signature of 11163. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 8 3     j = 7               9 5       j = 5             10 8         j = 3           9 9           j = 1         7 11             j = -1       5 8               j = -3     2 7                 j = -5   1 5                   j = -7   2                     j = -9 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, 163]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 163]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[14, 8, 15, 7], > X[16, 10, 17, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[8, 16, 9, 15], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 163]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -8, 5, -2, 6, -10, 7, -4, 8, -5, 9, -3, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 163]]
Out[5]=	DTCode[4, 10, 18, 14, 16, 2, 20, 8, 22, 12, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 163]][t]
Out[6]=	-4 6 15 24 2 3 4 -27 - t + -- - -- + -- + 24 t - 15 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 163]][z]
Out[7]=	2 4 6 8 1 + 2 z + z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 66], Knot[11, Alternating, 163]}
In[9]:=	{KnotDet[Knot[11, Alternating, 163]], KnotSignature[Knot[11, Alternating, 163]]}
Out[9]=	{119, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 163]][q]
Out[10]=	-4 3 7 12 2 3 4 5 6 7 -15 - q + -- - -- + -- + 19 q - 19 q + 17 q - 13 q + 8 q - 4 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 163]}
In[12]:=	A2Invariant[Knot[11, Alternating, 163]][q]
Out[12]=	-12 2 3 -2 2 4 6 8 10 12 14 3 - q - -- + -- - q + 3 q - 2 q + 4 q - 4 q + 2 q - q - 2 q + 6 4 q q 16 18 20 > 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 163]][a, z]
Out[13]=	2 2 4 4 2 2 2 2 z 7 z 2 2 4 3 z 9 z 2 4 5 - -- - 2 a + 10 z + ---- - ---- - 3 a z + 8 z + ---- - ---- - a z + 2 4 2 4 2 a a a a a 6 6 8 6 z 5 z z > 2 z + -- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 163]][a, z]
Out[14]=	2 2 2 2 2 2 z 4 z 3 2 z 2 z 14 z 5 + -- + 2 a - --- - --- - 4 a z - 2 a z - 18 z + -- - ---- - ----- - 2 3 a 6 4 2 a a a a a 3 3 3 3 4 2 2 2 z 5 z 12 z 6 z 3 3 3 4 z > 7 a z - ---- + ---- + ----- + ---- + 6 a z + 5 a z + 37 z + -- - 7 5 3 a 8 a a a a 4 4 4 5 5 5 5 7 z 8 z 40 z 2 4 4 z 14 z 13 z 13 z 5 > ---- + ---- + ----- + 13 a z + ---- - ----- - ----- + ----- + 4 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 3 5 6 8 z 17 z 44 z 2 6 11 z 5 z 28 z > 4 a z - 30 z + ---- - ----- - ----- - 11 a z + ----- - ---- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 11 z 12 z 2 8 7 z 11 z 9 > 11 a z + a z + 4 z + ----- + ----- + 3 a z + ---- + ----- + 4 a z + 4 2 3 a a a a 10 10 2 z > 2 z + ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 163]], Vassiliev[3][Knot[11, Alternating, 163]]}
Out[15]=	{2, 0}
In[16]:=	Kh[Knot[11, Alternating, 163]][q, t]
Out[16]=	3 1 2 1 5 2 7 5 8 11 q + 9 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 7 q 3 5 5 2 7 2 7 3 9 3 9 4 > --- + 9 q t + 10 q t + 8 q t + 9 q t + 5 q t + 8 q t + 3 q t + t 11 4 11 5 13 5 15 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a163

K11a162 K11a164