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

K11n167 K11n169

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,4,11,3 X14,6,15,5 X20,8,21,7 X4,10,5,9 X11,18,12,19 X2,14,3,13 X22,15,1,16 X8,18,9,17 X19,12,20,13 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 6t-2 + 18t-1 - 25 + 18t - 6t2 + t3

Conway Polynomial:

1 + 3z2 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{75, 2}

Jones Polynomial:

q-3 - 4q-2 + 7q-1 - 10 + 13q - 12q2 + 12q3 - 9q4 + 5q5 - 2q6

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

{K11n148, ...}

A2 (sl(3)) Invariant:

q-10 - q-8 - 2q-6 + 2q-4 - 2q-2 + 1 + 2q2 - q4 + 4q6 - q8 + 3q10 - 2q14 + 2q16 - 2q18 - q20

HOMFLY-PT Polynomial:

- a-6 - a-4z4 + 3a-2 + 5a-2z2 + 3a-2z4 + a-2z6 - 1 - 3z2 - 2z4 + a2z2

Kauffman Polynomial:

- 2a-7z + 3a-7z3 + a-6 - 2a-6z2 + 4a-6z4 + a-6z6 + a-5z3 + 3a-5z7 - a-4z2 + 3a-4z4 - 4a-4z6 + 4a-4z8 + 4a-3z - 3a-3z3 - 7a-3z5 + 3a-3z7 + 2a-3z9 - 3a-2 + a-2z2 + 6a-2z4 - 18a-2z6 + 9a-2z8 + 3a-1z + 6a-1z3 - 18a-1z5 + 4a-1z7 + 2a-1z9 - 1 + z2 + 5z4 - 12z6 + 5z8 + az + 7az3 - 11az5 + 4az7 + a2z2 - 2a2z4 + a2z6

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

{3, 4}

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 11168. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   2 j = 11                 3   j = 9               6 2   j = 7             6 3     j = 5           6 6       j = 3         7 6         j = 1       4 7           j = -1     3 6             j = -3   1 4               j = -5   3                 j = -7 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, 168]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 168]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 8, 21, 7], > X[4, 10, 5, 9], X[11, 18, 12, 19], X[2, 14, 3, 13], X[22, 15, 1, 16], > X[8, 18, 9, 17], X[19, 12, 20, 13], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 168]]
Out[4]=	GaussCode[1, -7, 2, -5, 3, -1, 4, -9, 5, -2, -6, 10, 7, -3, 8, -11, 9, 6, -10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 168]]
Out[5]=	DTCode[6, 10, 14, 20, 4, -18, 2, 22, 8, -12, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 168]][t]
Out[6]=	-3 6 18 2 3 -25 + t - -- + -- + 18 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 168]][z]
Out[7]=	2 6 1 + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 168]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 168]], KnotSignature[Knot[11, NonAlternating, 168]]}
Out[9]=	{75, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 168]][q]
Out[10]=	-3 4 7 2 3 4 5 6 -10 + q - -- + - + 13 q - 12 q + 12 q - 9 q + 5 q - 2 q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 148], Knot[11, NonAlternating, 168]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 168]][q]
Out[12]=	-10 -8 2 2 2 2 4 6 8 10 14 16 1 + q - q - -- + -- - -- + 2 q - q + 4 q - q + 3 q - 2 q + 2 q - 6 4 2 q q q 18 20 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 168]][a, z]
Out[13]=	2 4 4 6 -6 3 2 5 z 2 2 4 z 3 z z -1 - a + -- - 3 z + ---- + a z - 2 z - -- + ---- + -- 2 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 168]][a, z]
Out[14]=	2 2 2 3 -6 3 2 z 4 z 3 z 2 2 z z z 2 2 3 z -1 + a - -- - --- + --- + --- + a z + z - ---- - -- + -- + a z + ---- + 2 7 3 a 6 4 2 7 a a a a a a a 3 3 3 4 4 4 5 z 3 z 6 z 3 4 4 z 3 z 6 z 2 4 7 z > -- - ---- + ---- + 7 a z + 5 z + ---- + ---- + ---- - 2 a z - ---- - 5 3 a 6 4 2 3 a a a a a a 5 6 6 6 7 7 7 18 z 5 6 z 4 z 18 z 2 6 3 z 3 z 4 z > ----- - 11 a z - 12 z + -- - ---- - ----- + a z + ---- + ---- + ---- + a 6 4 2 5 3 a a a a a a 8 8 9 9 7 8 4 z 9 z 2 z 2 z > 4 a z + 5 z + ---- + ---- + ---- + ---- 4 2 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 168]], Vassiliev[3][Knot[11, NonAlternating, 168]]}
Out[15]=	{3, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 168]][q, t]
Out[16]=	3 1 3 1 4 3 6 4 q 3 7 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 6 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 6 q t + 6 q t + 6 q t + 3 q t + 6 q t + 2 q t + 3 q t + 13 5 > 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n168

K11n167 K11n169