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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X11,19,12,18 X17,9,18,22 X13,21,14,20 X19,13,20,12 X21,15,22,14 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

3q2 - 3q3 + 7q4 - 7q5 + 10q6 - 8q7 + 6q8 - 5q9 + 2q10 - q11

A2 (sl(3)) Invariant:

3q6 + 2q8 + 6q10 + 6q12 + 7q14 + 10q16 + 5q18 + 5q20 - q22 - 3q24 - 3q26 - 5q28 - 2q30 - 2q32 - q34

HOMFLY-PT Polynomial:

- 2a-10z-2 - 3a-10 - a-10z2 + 7a-8z-2 + 15a-8 + 12a-8z2 + 3a-8z4 - 8a-6z-2 - 22a-6 - 23a-6z2 - 11a-6z4 - 2a-6z6 + 3a-4z-2 + 10a-4 + 11a-4z2 + 3a-4z4

Kauffman Polynomial:

- a-13z-1 + 3a-13z - 3a-13z3 + a-13z5 + a-12 - 4a-12z4 + 2a-12z6 - a-11z-1 + 3a-11z - 6a-11z3 - a-11z5 + 2a-11z7 - 2a-10z-2 + 7a-10 - 9a-10z2 + 4a-10z4 - 3a-10z6 + 2a-10z8 + 7a-9z-1 - 21a-9z + 26a-9z3 - 13a-9z5 + 2a-9z7 + a-9z9 - 7a-8z-2 + 22a-8 - 36a-8z2 + 44a-8z4 - 24a-8z6 + 6a-8z8 + 15a-7z-1 - 45a-7z + 52a-7z3 - 23a-7z5 + 3a-7z7 + a-7z9 - 8a-6z-2 + 28a-6 - 46a-6z2 + 42a-6z4 - 19a-6z6 + 4a-6z8 + 8a-5z-1 - 24a-5z + 23a-5z3 - 12a-5z5 + 3a-5z7 - 3a-4z-2 + 13a-4 - 19a-4z2 + 6a-4z4

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                   1 j = 21                 1   j = 19               4 1   j = 17             2 1     j = 15           6 4       j = 13         4 2         j = 11       4 7           j = 9     3 3             j = 7     4               j = 5 3 3                 j = 3 3

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, 268]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 268]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 17, 8, 16], X[15, 5, 16, 8], > X[11, 19, 12, 18], X[17, 9, 18, 22], X[13, 21, 14, 20], X[19, 13, 20, 12], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 4}, > {-11, 2, -5, 8, -7, 9, -4, 3, -6, 5, -8, 7, -9, 6}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 10 11 3 q - 3 q + 7 q - 7 q + 10 q - 8 q + 6 q - 5 q + 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	6 8 10 12 14 16 18 20 22 24 3 q + 2 q + 6 q + 6 q + 7 q + 10 q + 5 q + 5 q - q - 3 q - 26 28 30 32 34 > 3 q - 5 q - 2 q - 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 268]][a, z]
Out[8]=	2 2 2 -3 15 22 10 2 7 8 3 z 12 z 23 z --- + -- - -- + -- - ------ + ----- - ----- + ----- - --- + ----- - ----- + 10 8 6 4 10 2 8 2 6 2 4 2 10 8 6 a a a a a z a z a z a z a a a 2 4 4 4 6 11 z 3 z 11 z 3 z 2 z > ----- + ---- - ----- + ---- - ---- 4 8 6 4 6 a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 268]][a, z]
Out[9]=	-12 7 22 28 13 2 7 8 3 1 1 a + --- + -- + -- + -- - ------ - ----- - ----- - ----- - ----- - ----- + 10 8 6 4 10 2 8 2 6 2 4 2 13 11 a a a a a z a z a z a z a z a z 2 2 7 15 8 3 z 3 z 21 z 45 z 24 z 9 z 36 z > ---- + ---- + ---- + --- + --- - ---- - ---- - ---- - ---- - ----- - 9 7 5 13 11 9 7 5 10 8 a z a z a z a a a a a a a 2 2 3 3 3 3 3 4 4 4 46 z 19 z 3 z 6 z 26 z 52 z 23 z 4 z 4 z 44 z > ----- - ----- - ---- - ---- + ----- + ----- + ----- - ---- + ---- + ----- + 6 4 13 11 9 7 5 12 10 8 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 42 z 6 z z z 13 z 23 z 12 z 2 z 3 z 24 z > ----- + ---- + --- - --- - ----- - ----- - ----- + ---- - ---- - ----- - 6 4 13 11 9 7 5 12 10 8 a a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 19 z 2 z 2 z 3 z 3 z 2 z 6 z 4 z z z > ----- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 6 11 9 7 5 10 8 6 9 7 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 5 7 2 9 2 9 3 11 3 11 4 3 q + 3 q + 3 q t + 4 q t + 3 q t + 3 q t + 4 q t + 7 q t + 13 4 13 5 15 5 15 6 17 6 17 7 19 7 > 4 q t + 2 q t + 6 q t + 4 q t + 2 q t + q t + 4 q t + 19 8 21 8 23 9 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n268

L11n267 L11n269