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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- 2q-6 + 6q-5 - 8q-4 + 12q-3 - 12q-2 + 12q-1 - 9 + 7q - 3q2 + q3

A2 (sl(3)) Invariant:

- q-22 - 2q-18 + 3q-16 + 2q-14 + 4q-12 + 6q-10 + 5q-6 - q-4 + 4q-2 + 3 + q2 + 3q4 - q6 + q8

HOMFLY-PT Polynomial:

z-2 + 5 + 6z2 + 4z4 + z6 - 2a2z-2 - 11a2 - 17a2z2 - 14a2z4 - 6a2z6 - a2z8 + a4z-2 + 8a4 + 10a4z2 + 5a4z4 + a4z6 - 2a6 - a6z2

Kauffman Polynomial:

- a-2 + 3a-2z2 - 3a-2z4 + a-2z6 - a-1z + 5a-1z3 - 8a-1z5 + 3a-1z7 - z-2 + 4 - 3z2 + 3z4 - 9z6 + 4z8 + 2az-1 - 10az + 21az3 - 21az5 + 3az7 + 2az9 - 2a2z-2 + 12a2 - 26a2z2 + 33a2z4 - 29a2z6 + 10a2z8 + 2a3z-1 - 14a3z + 30a3z3 - 24a3z5 + 5a3z7 + 2a3z9 - a4z-2 + 10a4 - 28a4z2 + 33a4z4 - 18a4z6 + 6a4z8 - 7a5z + 17a5z3 - 11a5z5 + 5a5z7 + 2a6 - 8a6z2 + 6a6z4 + a6z6 - 2a7z + 3a7z3

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                   1 j = 5                 2   j = 3               5 1   j = 1             5 3     j = -1           7 4       j = -3         6 6         j = -5       6 6           j = -7     3 7             j = -9   3 5               j = -11   4                 j = -13 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, 371]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 371]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 10, 4, 11], X[7, 14, 8, 15], X[15, 22, 16, 17], > X[17, 16, 18, 5], X[9, 19, 10, 18], X[13, 21, 14, 20], X[19, 13, 20, 12], > X[21, 9, 22, 8], X[2, 5, 3, 6], X[11, 4, 12, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {-5, 6, -8, 7, -9, 4}, > {10, -1, -3, 9, -6, 2, -11, 8, -7, 3, -4, 5}]
In[6]:=	Jones[L][q]
Out[6]=	2 6 8 12 12 12 2 3 -9 - -- + -- - -- + -- - -- + -- + 7 q - 3 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 3 2 4 6 5 -4 4 2 4 6 8 3 - q - --- + --- + --- + --- + --- + -- - q + -- + q + 3 q - q + q 18 16 14 12 10 6 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 371]][a, z]
Out[8]=	2 4 2 4 6 -2 2 a a 2 2 2 4 2 5 - 11 a + 8 a - 2 a + z - ---- + -- + 6 z - 17 a z + 10 a z - 2 2 z z 6 2 4 2 4 4 4 6 2 6 4 6 2 8 > a z + 4 z - 14 a z + 5 a z + z - 6 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 371]][a, z]
Out[9]=	2 4 3 -2 2 4 6 -2 2 a a 2 a 2 a z 4 - a + 12 a + 10 a + 2 a - z - ---- - -- + --- + ---- - - - 10 a z - 2 2 z z a z z 2 3 5 7 2 3 z 2 2 4 2 6 2 > 14 a z - 7 a z - 2 a z - 3 z + ---- - 26 a z - 28 a z - 8 a z + 2 a 3 4 5 z 3 3 3 5 3 7 3 4 3 z 2 4 > ---- + 21 a z + 30 a z + 17 a z + 3 a z + 3 z - ---- + 33 a z + a 2 a 5 6 4 4 6 4 8 z 5 3 5 5 5 6 z > 33 a z + 6 a z - ---- - 21 a z - 24 a z - 11 a z - 9 z + -- - a 2 a 7 2 6 4 6 6 6 3 z 7 3 7 5 7 8 > 29 a z - 18 a z + a z + ---- + 3 a z + 5 a z + 5 a z + 4 z + a 2 8 4 8 9 3 9 > 10 a z + 6 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 7 2 4 3 5 3 7 6 6 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 6 4 t 2 3 2 3 3 5 3 7 4 > ---- + --- + 5 q t + 3 q t + 5 q t + q t + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n371

L11n370 L11n372