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L11n369 L11n371

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,8,15,7 X15,20,16,21 X11,19,12,18 X17,13,18,12 X19,22,20,17 X21,16,22,5 X8,14,9,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-6 + 5q-5 - 7q-4 + 11q-3 - 10q-2 + 11q-1 - 9 + 6q - 3q2 + q3

A2 (sl(3)) Invariant:

5q-16 + 3q-14 + 5q-12 + 7q-10 + 2q-8 + 5q-6 - q-4 + q-2 - 2q2 + 2q4 - q6 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 - 2 - 4z2 - 2z4 + a2z-2 + 3a2 + 4a2z2 + 3a2z4 + a2z6 - 2a4z-2 - 2a4 - a4z2 - a4z4 + a6z-2

Kauffman Polynomial:

- a-2 + 3a-2z2 - 3a-2z4 + a-2z6 - 2a-1z + 8a-1z3 - 9a-1z5 + 3a-1z7 - 1 + 8z2 - 7z4 - 4z6 + 3z8 - 6az + 19az3 - 25az5 + 7az7 + az9 + a2z-2 - a2 + 3a2z2 - 3a2z4 - 9a2z6 + 6a2z8 - 2a3z-1 - 2a3z + 12a3z3 - 16a3z5 + 6a3z7 + a3z9 + 2a4z-2 - 2a4 - 3a4z2 + 6a4z4 - 4a4z6 + 3a4z8 - 2a5z-1 + 2a5z + 2a5z3 + 2a5z7 + a6z-2 - 2a6 - a6z2 + 5a6z4 + a7z3

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               4 1   j = 1             5 2     j = -1           6 4       j = -3         6 7         j = -5       5 4           j = -7     2 6             j = -9   3 5               j = -11   4                 j = -13 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]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 370]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 370]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[15, 20, 16, 21], > X[11, 19, 12, 18], X[17, 13, 18, 12], X[19, 22, 20, 17], X[21, 16, 22, 5], > X[8, 14, 9, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-6, 5, -7, 4, -8, 7}, > {10, -1, 3, -9, 11, -2, -5, 6, 9, -3, -4, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-6 5 7 11 10 11 2 3 -9 - q + -- - -- + -- - -- + -- + 6 q - 3 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	5 3 5 7 2 5 -4 -2 2 4 6 10 --- + --- + --- + --- + -- + -- - q + q - 2 q + 2 q - q + q 16 14 12 10 8 6 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 370]][a, z]
Out[8]=	2 4 6 2 -2 2 4 a 2 a a 2 z 2 2 4 2 4 -2 + a + 3 a - 2 a + -- - ---- + -- - 4 z + -- + 4 a z - a z - 2 z + 2 2 2 2 z z z a 2 4 4 4 2 6 > 3 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 370]][a, z]
Out[9]=	2 4 6 3 5 -2 2 4 6 a 2 a a 2 a 2 a 2 z -1 - a - a - 2 a - 2 a + -- + ---- + -- - ---- - ---- - --- - 6 a z - 2 2 2 z z a z z z 2 3 3 5 2 3 z 2 2 4 2 6 2 8 z > 2 a z + 2 a z + 8 z + ---- + 3 a z - 3 a z - a z + ---- + 2 a a 4 3 3 3 5 3 7 3 4 3 z 2 4 4 4 > 19 a z + 12 a z + 2 a z + a z - 7 z - ---- - 3 a z + 6 a z + 2 a 5 6 6 4 9 z 5 3 5 6 z 2 6 4 6 > 5 a z - ---- - 25 a z - 16 a z - 4 z + -- - 9 a z - 4 a z + a 2 a 7 3 z 7 3 7 5 7 8 2 8 4 8 9 3 9 > ---- + 7 a z + 6 a z + 2 a z + 3 z + 6 a z + 3 a z + a z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	7 6 1 4 3 5 2 6 5 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 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 + 2 q t + 4 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 L11n370

L11n369 L11n371