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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X22,18,19,17 X20,12,21,11 X12,20,13,19 X18,22,5,21 X9,16,10,17 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- 2q-3 + 4q-2 - 5q-1 + 6 - 5q + 6q2 - 2q3 + 2q4

A2 (sl(3)) Invariant:

- 2q-10 - q-6 - 3q-4 - 1 + 4q2 + 5q4 + 7q6 + 8q8 + 4q10 + 4q12 + 2q14

HOMFLY-PT Polynomial:

2a-4z-2 + 2a-4 - 5a-2z-2 - 8a-2 - 4a-2z2 + 4z-2 + 8 + 6z2 + 2z4 - a2z-2 - 2a2 - 2a2z2

Kauffman Polynomial:

- 2a-4z-2 + 8a-4 - 9a-4z2 + 3a-4z4 + 5a-3z-1 - 13a-3z + 9a-3z3 - 3a-3z5 + a-3z7 - 5a-2z-2 + 16a-2 - 20a-2z2 + 11a-2z4 - 3a-2z6 + a-2z8 + 9a-1z-1 - 24a-1z + 19a-1z3 - 8a-1z5 + 3a-1z7 - 4z-2 + 11 - 15z2 + 10z4 - 2z6 + z8 + 5az-1 - 14az + 13az3 - 5az5 + 2az7 - a2z-2 + 2a2 - 4a2z2 + 2a2z4 + a2z6 + a3z-1 - 3a3z + 3a3z3

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9               2 j = 7             2 2 j = 5           4     j = 3         1 2     j = 1       5 4       j = -1     2 3         j = -3   2 3           j = -5   2             j = -7 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]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 382]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 382]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], > X[8, 4, 9, 3], X[22, 18, 19, 17], X[20, 12, 21, 11], X[12, 20, 13, 19], > X[18, 22, 5, 21], X[9, 16, 10, 17], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {8, -7, 9, -6}, > {-4, -1, 2, -5, -10, 4, 7, -8, 11, -2, 3, 10, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 5 2 3 4 6 - -- + -- - - - 5 q + 6 q - 2 q + 2 q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -6 3 2 4 6 8 10 12 14 -1 - --- - q - -- + 4 q + 5 q + 7 q + 8 q + 4 q + 4 q + 2 q 10 4 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 382]][a, z]
Out[8]=	2 2 2 8 2 4 2 5 a 2 4 z 2 2 4 8 + -- - -- - 2 a + -- + ----- - ----- - -- + 6 z - ---- - 2 a z + 2 z 4 2 2 4 2 2 2 2 2 a a z a z a z z a
In[9]:=	Kauffman[Link[11, NonAlternating, 382]][a, z]
Out[9]=	2 3 8 16 2 4 2 5 a 5 9 5 a a 13 z 11 + -- + -- + 2 a - -- - ----- - ----- - -- + ---- + --- + --- + -- - ---- - 4 2 2 4 2 2 2 2 3 a z z z 3 a a z a z a z z a z a 2 2 3 3 24 z 3 2 9 z 20 z 2 2 9 z 19 z > ---- - 14 a z - 3 a z - 15 z - ---- - ----- - 4 a z + ---- + ----- + a 4 2 3 a a a a 4 4 5 5 3 3 3 4 3 z 11 z 2 4 3 z 8 z 5 > 13 a z + 3 a z + 10 z + ---- + ----- + 2 a z - ---- - ---- - 5 a z - 4 2 3 a a a a 6 7 7 8 6 3 z 2 6 z 3 z 7 8 z > 2 z - ---- + a z + -- + ---- + 2 a z + z + -- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 2 2 3 2 3 3 2 - + 5 q + ----- + ----- + ----- + ---- + --- + 4 q t + q t + 2 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 7 3 7 4 9 4 > 4 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n382

L11n381 L11n383