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

K11n159 K11n161

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X12,6,13,5 X20,8,21,7 X16,10,17,9 X11,19,12,18 X22,13,1,14 X8,16,9,15 X17,4,18,5 X2,19,3,20 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 6t-2 + 16t-1 - 21 + 16t - 6t2 + t3

Conway Polynomial:

1 + z2 + z6

Other knots with the same Alexander/Conway Polynomial:

{K11n7, K11n131, ...}

Determinant and Signature:

{67, 2}

Jones Polynomial:

2q-1 - 5 + 8q - 10q2 + 12q3 - 11q4 + 9q5 - 6q6 + 3q7 - q8

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

{...}

A2 (sl(3)) Invariant:

2q-4 - 1 + 2q2 - 3q4 + q6 + 3q12 - q14 + 3q16 - q18 - 2q20 + q22 - q24

HOMFLY-PT Polynomial:

- 2a-6 - 2a-6z2 - a-6z4 + 5a-4 + 8a-4z2 + 4a-4z4 + a-4z6 - 4a-2 - 7a-2z2 - 3a-2z4 + 2 + 2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 3a-8z6 - 3a-7z + 9a-7z3 - 12a-7z5 + 5a-7z7 + 2a-6 - 8a-6z2 + 15a-6z4 - 13a-6z6 + 5a-6z8 - 4a-5z + 14a-5z3 - 10a-5z5 + a-5z7 + 2a-5z9 + 5a-4 - 21a-4z2 + 35a-4z4 - 24a-4z6 + 8a-4z8 - 4a-3z3 + 6a-3z5 - 3a-3z7 + 2a-3z9 + 4a-2 - 17a-2z2 + 17a-2z4 - 8a-2z6 + 3a-2z8 + a-1z - 7a-1z3 + 3a-1z5 + a-1z7 + 2 - 5z2 + 3z4

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

{1, 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 11160. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

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, 160]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 160]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[16, 10, 17, 9], X[11, 19, 12, 18], X[22, 13, 1, 14], X[8, 16, 9, 15], > X[17, 4, 18, 5], X[2, 19, 3, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 160]]
Out[4]=	GaussCode[1, -10, -2, 9, 3, -1, 4, -8, 5, 2, -6, -3, 7, -11, 8, -5, -9, 6, 10, > -4, 11, -7]
In[5]:=	DTCode[Knot[11, NonAlternating, 160]]
Out[5]=	DTCode[6, -10, 12, 20, 16, -18, 22, 8, -4, 2, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 160]][t]
Out[6]=	-3 6 16 2 3 -21 + t - -- + -- + 16 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 160]][z]
Out[7]=	2 6 1 + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 7], Knot[11, NonAlternating, 131], > Knot[11, NonAlternating, 160]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 160]], KnotSignature[Knot[11, NonAlternating, 160]]}
Out[9]=	{67, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 160]][q]
Out[10]=	2 2 3 4 5 6 7 8 -5 + - + 8 q - 10 q + 12 q - 11 q + 9 q - 6 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 160]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 160]][q]
Out[12]=	2 2 4 6 12 14 16 18 20 22 24 -1 + -- + 2 q - 3 q + q + 3 q - q + 3 q - q - 2 q + q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 160]][a, z]
Out[13]=	2 2 2 4 4 4 6 2 5 4 2 2 z 8 z 7 z z 4 z 3 z z 2 - -- + -- - -- + 2 z - ---- + ---- - ---- - -- + ---- - ---- + -- 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 160]][a, z]
Out[14]=	2 2 2 2 3 2 5 4 3 z 4 z z 2 z 8 z 21 z 17 z 2 z 2 + -- + -- + -- - --- - --- + - - 5 z + -- - ---- - ----- - ----- - ---- + 6 4 2 7 5 a 8 6 4 2 9 a a a a a a a a a a 3 3 3 3 4 4 4 4 5 9 z 14 z 4 z 7 z 4 6 z 15 z 35 z 17 z z > ---- + ----- - ---- - ---- + 3 z - ---- + ----- + ----- + ----- + -- - 7 5 3 a 8 6 4 2 9 a a a a a a a a 5 5 5 5 6 6 6 6 7 7 12 z 10 z 6 z 3 z 3 z 13 z 24 z 8 z 5 z z > ----- - ----- + ---- + ---- + ---- - ----- - ----- - ---- + ---- + -- - 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a 7 7 8 8 8 9 9 3 z z 5 z 8 z 3 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, 160]], Vassiliev[3][Knot[11, NonAlternating, 160]]}
Out[15]=	{1, 3}
In[16]:=	Kh[Knot[11, NonAlternating, 160]][q, t]
Out[16]=	3 2 3 2 q 3 5 5 2 7 2 5 q + 4 q + ----- + --- + --- + 6 q t + 4 q t + 6 q t + 6 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 > 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 15 6 17 7 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n160

K11n159 K11n161