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

K11n162 K11n164

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X5,12,6,13 X22,8,1,7 X9,16,10,17 X11,19,12,18 X8,14,9,13 X20,16,21,15 X17,4,18,5 X19,3,20,2 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 -10 -12 22 -16 -18 8 20 -4 -2 14

Alexander Polynomial:

t-3 - 8t-2 + 22t-1 - 29 + 22t - 8t2 + t3

Conway Polynomial:

1 - z2 - 2z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{10105, ...}

Determinant and Signature:

{91, 2}

Jones Polynomial:

- q-2 + 5q-1 - 9 + 13q - 15q2 + 16q3 - 14q4 + 10q5 - 6q6 + 2q7

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

{...}

A2 (sl(3)) Invariant:

- q-6 + 3q-4 - q-2 + 4q2 - 3q4 + 3q6 - 2q8 + q12 - 3q14 + 3q16 - 2q18 - q20 + 2q22

HOMFLY-PT Polynomial:

a-6 + 2a-6z2 - 2a-4 - 5a-4z2 - 3a-4z4 + a-2 + 2a-2z2 + 2a-2z4 + a-2z6 + 1 - z4

Kauffman Polynomial:

- 3a-8z2 + 3a-8z4 + 2a-7z - 7a-7z3 + 5a-7z5 + a-7z7 - a-6 + a-6z2 + a-6z4 - a-6z6 + 3a-6z8 + 3a-5z - 2a-5z3 - 5a-5z5 + 4a-5z7 + 2a-5z9 - 2a-4 + 9a-4z2 - 5a-4z4 - 11a-4z6 + 10a-4z8 + a-3z + 9a-3z3 - 26a-3z5 + 12a-3z7 + 2a-3z9 - a-2 + 5a-2z2 - 9a-2z4 - 5a-2z6 + 7a-2z8 + 4a-1z3 - 15a-1z5 + 9a-1z7 + 1 - 6z4 + 5z6 + az5

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

{-1, -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=2 is the signature of 11163. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                   2 j = 13                 4   j = 11               6 2   j = 9             8 4     j = 7           8 6       j = 5         7 8         j = 3       6 8           j = 1     4 8             j = -1   1 5               j = -3   4                 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, NonAlternating, 163]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 163]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[5, 12, 6, 13], X[22, 8, 1, 7], > X[9, 16, 10, 17], X[11, 19, 12, 18], X[8, 14, 9, 13], X[20, 16, 21, 15], > X[17, 4, 18, 5], X[19, 3, 20, 2], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 163]]
Out[4]=	GaussCode[1, 10, -2, 9, -3, -1, 4, -7, -5, 2, -6, 3, 7, -11, 8, 5, -9, 6, -10, > -8, 11, -4]
In[5]:=	DTCode[Knot[11, NonAlternating, 163]]
Out[5]=	DTCode[6, -10, -12, 22, -16, -18, 8, 20, -4, -2, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 163]][t]
Out[6]=	-3 8 22 2 3 -29 + t - -- + -- + 22 t - 8 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 163]][z]
Out[7]=	2 4 6 1 - z - 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 105], Knot[11, NonAlternating, 163]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 163]], KnotSignature[Knot[11, NonAlternating, 163]]}
Out[9]=	{91, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 163]][q]
Out[10]=	-2 5 2 3 4 5 6 7 -9 - q + - + 13 q - 15 q + 16 q - 14 q + 10 q - 6 q + 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 163]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 163]][q]
Out[12]=	-6 3 -2 2 4 6 8 12 14 16 18 -q + -- - q + 4 q - 3 q + 3 q - 2 q + q - 3 q + 3 q - 2 q - 4 q 20 22 > q + 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 163]][a, z]
Out[13]=	2 2 2 4 4 6 -6 2 -2 2 z 5 z 2 z 4 3 z 2 z z 1 + a - -- + a + ---- - ---- + ---- - z - ---- + ---- + -- 4 6 4 2 4 2 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 163]][a, z]
Out[14]=	2 2 2 2 3 3 -6 2 -2 2 z 3 z z 3 z z 9 z 5 z 7 z 2 z 1 - a - -- - a + --- + --- + -- - ---- + -- + ---- + ---- - ---- - ---- + 4 7 5 3 8 6 4 2 7 5 a a a a a a a a a a 3 3 4 4 4 4 5 5 5 9 z 4 z 4 3 z z 5 z 9 z 5 z 5 z 26 z > ---- + ---- - 6 z + ---- + -- - ---- - ---- + ---- - ---- - ----- - 3 a 8 6 4 2 7 5 3 a a a a a a a a 5 6 6 6 7 7 7 7 8 15 z 5 6 z 11 z 5 z z 4 z 12 z 9 z 3 z > ----- + a z + 5 z - -- - ----- - ---- + -- + ---- + ----- + ---- + ---- + a 6 4 2 7 5 3 a 6 a a a a a a a 8 8 9 9 10 z 7 z 2 z 2 z > ----- + ---- + ---- + ---- 4 2 5 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 163]], Vassiliev[3][Knot[11, NonAlternating, 163]]}
Out[15]=	{-1, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 163]][q, t]
Out[16]=	3 1 4 1 5 4 q 3 5 5 2 8 q + 6 q + ----- + ----- + ---- + --- + --- + 8 q t + 7 q t + 8 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 8 q t + 6 q t + 8 q t + 4 q t + 6 q t + 2 q t + 4 q t + 15 6 > 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n163

K11n162 K11n164