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

K11n160 K11n162

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X12,6,13,5 X20,8,21,7 X18,10,19,9 X16,11,17,12 X13,1,14,22 X4,16,5,15 X2,17,3,18 X8,20,9,19 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-3 - 8t-2 + 14t-1 - 15 + 14t - 8t2 + 2t3

Conway Polynomial:

1 + 4z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{10108, ...}

Determinant and Signature:

{63, 2}

Jones Polynomial:

- q-2 + 3q-1 - 5 + 9q - 10q2 + 11q3 - 10q4 + 7q5 - 5q6 + 2q7

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

{...}

A2 (sl(3)) Invariant:

- q-6 + q-4 + 3q2 - q4 + 3q6 + q12 - 3q14 + q16 - 2q18 - q20 + q22 + q26

HOMFLY-PT Polynomial:

a-8 - 2a-6 - 3a-6z2 - a-6z4 + 2a-4z2 + 3a-4z4 + a-4z6 + 2a-2 + 3a-2z2 + 3a-2z4 + a-2z6 - 2z2 - z4

Kauffman Polynomial:

a-8 - 5a-8z2 + 3a-8z4 + 4a-7z - 9a-7z3 + 3a-7z5 + a-7z7 + 2a-6 - 15a-6z2 + 18a-6z4 - 9a-6z6 + 3a-6z8 + 8a-5z - 17a-5z3 + 14a-5z5 - 5a-5z7 + 2a-5z9 - 14a-4z2 + 30a-4z4 - 21a-4z6 + 7a-4z8 + 4a-3z - 3a-3z3 + 2a-3z5 - 2a-3z7 + 2a-3z9 - 2a-2 - a-2z2 + 8a-2z4 - 9a-2z6 + 4a-2z8 + 3a-1z3 - 8a-1z5 + 4a-1z7 + 3z2 - 7z4 + 3z6 - 2az3 + az5

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=2 is the signature of 11161. 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                 3   j = 11               4 2   j = 9             6 3     j = 7           5 4       j = 5         5 6         j = 3       4 5           j = 1     2 6             j = -1   1 3               j = -3   2                 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, 161]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 161]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[18, 10, 19, 9], X[16, 11, 17, 12], X[13, 1, 14, 22], X[4, 16, 5, 15], > X[2, 17, 3, 18], X[8, 20, 9, 19], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 161]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -1, 4, -10, 5, -2, 6, -3, -7, 11, 8, -6, 9, -5, 10, > -4, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 161]]
Out[5]=	DTCode[6, 10, 12, 20, 18, 16, -22, 4, 2, 8, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 161]][t]
Out[6]=	2 8 14 2 3 -15 + -- - -- + -- + 14 t - 8 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 161]][z]
Out[7]=	4 6 1 + 4 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 108], Knot[11, NonAlternating, 161]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 161]], KnotSignature[Knot[11, NonAlternating, 161]]}
Out[9]=	{63, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 161]][q]
Out[10]=	-2 3 2 3 4 5 6 7 -5 - q + - + 9 q - 10 q + 11 q - 10 q + 7 q - 5 q + 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 161]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 161]][q]
Out[12]=	-6 -4 2 4 6 12 14 16 18 20 22 26 -q + q + 3 q - q + 3 q + q - 3 q + q - 2 q - q + q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 161]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -8 2 2 2 3 z 2 z 3 z 4 z 3 z 3 z z z a - -- + -- - 2 z - ---- + ---- + ---- - z - -- + ---- + ---- + -- + -- 6 2 6 4 2 6 4 2 4 2 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 161]][a, z]
Out[14]=	2 2 2 2 3 -8 2 2 4 z 8 z 4 z 2 5 z 15 z 14 z z 9 z a + -- - -- + --- + --- + --- + 3 z - ---- - ----- - ----- - -- - ---- - 6 2 7 5 3 8 6 4 2 7 a a a a a a a a a a 3 3 3 4 4 4 4 5 17 z 3 z 3 z 3 4 3 z 18 z 30 z 8 z 3 z > ----- - ---- + ---- - 2 a z - 7 z + ---- + ----- + ----- + ---- + ---- + 5 3 a 8 6 4 2 7 a a a a a a a 5 5 5 6 6 6 7 7 14 z 2 z 8 z 5 6 9 z 21 z 9 z z 5 z > ----- + ---- - ---- + a z + 3 z - ---- - ----- - ---- + -- - ---- - 5 3 a 6 4 2 7 5 a a a a a a a 7 7 8 8 8 9 9 2 z 4 z 3 z 7 z 4 z 2 z 2 z > ---- + ---- + ---- + ---- + ---- + ---- + ---- 3 a 6 4 2 5 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 161]], Vassiliev[3][Knot[11, NonAlternating, 161]]}
Out[15]=	{0, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 161]][q, t]
Out[16]=	3 1 2 1 3 2 q 3 5 5 2 6 q + 4 q + ----- + ----- + ---- + --- + --- + 5 q t + 5 q t + 6 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 > 5 q t + 4 q t + 6 q t + 3 q t + 4 q t + 2 q t + 3 q t + 15 6 > 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n161

K11n160 K11n162