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L11n357 L11n359

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,19,12,18 X7,16,8,17 X15,8,16,9 X17,15,18,22 X21,13,22,12 X13,21,14,20 X19,5,20,14 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-20 + q-18 - q-16 + q-14 + q-12 + q-10 + 5q-8 + 4q-6 + 5q-4 + 4q-2 + 2 + 3q2 - q4 + q6 + q8 - q10

HOMFLY-PT Polynomial:

- a-2z2 + z-2 + 2 + z2 + z4 - 2a2z-2 - 2a2 + a2z4 + a4z-2 - a4 - 2a4z2 + a6

Kauffman Polynomial:

- 2a-3z + a-3z3 + 2a-2 - 5a-2z2 + 3a-2z4 - 7a-1z + 15a-1z3 - 8a-1z5 + 2a-1z7 - z-2 + 10 - 26z2 + 32z4 - 15z6 + 3z8 + 2az-1 - 14az + 24az3 - 11az5 + az9 - 2a2z-2 + 12a2 - 30a2z2 + 37a2z4 - 22a2z6 + 5a2z8 + 2a3z-1 - 10a3z + 15a3z3 - 10a3z5 + a3z9 - a4z-2 + 4a4 - 5a4z2 + 4a4z4 - 6a4z6 + 2a4z8 - a5z + 5a5z3 - 7a5z5 + 2a5z7 - a6 + 4a6z2 - 4a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   1 j = 5                 2   j = 3               3 2   j = 1             4 1     j = -1           3 4       j = -3         4 3         j = -5       2 4           j = -7     2 3             j = -9   1 3               j = -11   1                 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, 358]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 358]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 19, 12, 18], X[7, 16, 8, 17], > X[15, 8, 16, 9], X[17, 15, 18, 22], X[21, 13, 22, 12], X[13, 21, 14, 20], > X[19, 5, 20, 14], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-5, 4, -6, 3, -9, 8, -7, 6}, > {10, -1, -4, 5, 11, -2, -3, 7, -8, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 4 5 7 6 2 3 7 + q - -- + -- - -- + -- - - - 4 q + 3 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 -14 -12 -10 5 4 5 4 2 4 2 + q + q - q + q + q + q + -- + -- + -- + -- + 3 q - q + 8 6 4 2 q q q q 6 8 10 > q + q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 358]][a, z]
Out[8]=	2 4 2 2 4 6 -2 2 a a 2 z 4 2 4 2 4 2 - 2 a - a + a + z - ---- + -- + z - -- - 2 a z + z + a z 2 2 2 z z a
In[9]:=	Kauffman[Link[11, NonAlternating, 358]][a, z]
Out[9]=	2 4 3 2 2 4 6 -2 2 a a 2 a 2 a 2 z 7 z 10 + -- + 12 a + 4 a - a - z - ---- - -- + --- + ---- - --- - --- - 2 2 2 z z 3 a a z z a 2 3 5 2 5 z 2 2 4 2 6 2 > 14 a z - 10 a z - a z - 26 z - ---- - 30 a z - 5 a z + 4 a z + 2 a 3 3 4 z 15 z 3 3 3 5 3 4 3 z 2 4 > -- + ----- + 24 a z + 15 a z + 5 a z + 32 z + ---- + 37 a z + 3 a 2 a a 5 4 4 6 4 8 z 5 3 5 5 5 6 > 4 a z - 4 a z - ---- - 11 a z - 10 a z - 7 a z - 15 z - a 7 2 6 4 6 6 6 2 z 5 7 8 2 8 4 8 > 22 a z - 6 a z + a z + ---- + 2 a z + 3 z + 5 a z + 2 a z + a 9 3 9 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 3 2 3 2 4 - + 4 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 4 3 3 3 3 2 5 2 7 3 > ----- + ---- + --- + q t + 3 q t + 2 q t + 2 q t + q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n358

L11n357 L11n359