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

K11n154 K11n156

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X12,6,13,5 X20,8,21,7 X18,9,19,10 X16,11,17,12 X13,1,14,22 X4,16,5,15 X10,17,11,18 X2,19,3,20 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-3 - 8t-2 + 11t-1 - 9 + 11t - 8t2 + 2t3

Conway Polynomial:

1 - 3z2 + 4z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{51, 2}

Jones Polynomial:

- q-4 + 3q-3 - 4q-2 + 7q-1 - 8 + 8q - 8q2 + 6q3 - 4q4 + 2q5

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

{...}

A2 (sl(3)) Invariant:

- q-12 + q-10 + q-8 + q-6 + 2q-4 - q-2 + 1 - q2 - q4 + q6 - 2q8 + q10 - q12 + q16 + q20

HOMFLY-PT Polynomial:

a-6 - a-4 - 3a-4z2 - a-4z4 + a-2z2 + 3a-2z4 + a-2z6 + z2 + 3z4 + z6 + a2 - 2a2z2 - a2z4

Kauffman Polynomial:

- a-6 + 3a-6z2 - a-5z + 3a-5z3 + a-5z5 - a-4 + 3a-4z2 - 3a-4z4 + 3a-4z6 + 3a-3z - a-3z3 - 6a-3z5 + 4a-3z7 - 6a-2z2 + 8a-2z4 - 10a-2z6 + 4a-2z8 + 6a-1z - 7a-1z3 - a-1z5 - 3a-1z7 + 2a-1z9 - 12z2 + 30z4 - 27z6 + 7z8 + 2az + az3 + 2az5 - 6az7 + 2az9 - a2 - 6a2z2 + 19a2z4 - 14a2z6 + 3a2z8 + 4a3z3 - 4a3z5 + a3z7

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

{-3, -3}

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 11155. 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 j = 11                   2 j = 9                 2   j = 7               4 2   j = 5             4 2     j = 3           4 4       j = 1         5 5         j = -1       2 3           j = -3     2 5             j = -5   1 2               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, NonAlternating, 155]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 155]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[18, 9, 19, 10], X[16, 11, 17, 12], X[13, 1, 14, 22], X[4, 16, 5, 15], > X[10, 17, 11, 18], X[2, 19, 3, 20], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 155]]
Out[4]=	GaussCode[1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, -7, 11, 8, -6, 9, -5, 10, > -4, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 155]]
Out[5]=	DTCode[6, 8, 12, 20, 18, 16, -22, 4, 10, 2, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 155]][t]
Out[6]=	2 8 11 2 3 -9 + -- - -- + -- + 11 t - 8 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 155]][z]
Out[7]=	2 4 6 1 - 3 z + 4 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 155]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 155]], KnotSignature[Knot[11, NonAlternating, 155]]}
Out[9]=	{51, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 155]][q]
Out[10]=	-4 3 4 7 2 3 4 5 -8 - q + -- - -- + - + 8 q - 8 q + 6 q - 4 q + 2 q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 155]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 155]][q]
Out[12]=	-12 -10 -8 -6 2 -2 2 4 6 8 10 12 1 - q + q + q + q + -- - q - q - q + q - 2 q + q - q + 4 q 16 20 > q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 155]][a, z]
Out[13]=	2 2 4 4 6 -6 -4 2 2 3 z z 2 2 4 z 3 z 2 4 6 z a - a + a + z - ---- + -- - 2 a z + 3 z - -- + ---- - a z + z + -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 155]][a, z]
Out[14]=	2 2 2 -6 -4 2 z 3 z 6 z 2 3 z 3 z 6 z -a - a - a - -- + --- + --- + 2 a z - 12 z + ---- + ---- - ---- - 5 3 a 6 4 2 a a a a a 3 3 3 4 4 2 2 3 z z 7 z 3 3 3 4 3 z 8 z > 6 a z + ---- - -- - ---- + a z + 4 a z + 30 z - ---- + ---- + 5 3 a 4 2 a a a a 5 5 5 6 6 2 4 z 6 z z 5 3 5 6 3 z 10 z > 19 a z + -- - ---- - -- + 2 a z - 4 a z - 27 z + ---- - ----- - 5 3 a 4 2 a a a a 7 7 8 9 2 6 4 z 3 z 7 3 7 8 4 z 2 8 2 z > 14 a z + ---- - ---- - 6 a z + a z + 7 z + ---- + 3 a z + ---- + 3 a 2 a a a 9 > 2 a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 155]], Vassiliev[3][Knot[11, NonAlternating, 155]]}
Out[15]=	{-3, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 155]][q, t]
Out[16]=	3 1 2 1 2 2 5 2 3 5 q 5 q + 4 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 11 4 > 4 q t + 4 q t + 2 q t + 4 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n155

K11n154 K11n156