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L11n55 L11n57

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,17,12,16 X14,7,15,8 X8,15,9,16 X13,21,14,20 X17,5,18,22 X21,19,22,18 X19,13,20,12 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-9/2 + 2q-7/2 - 5q-5/2 + 6q-3/2 - 8q-1/2 + 7q1/2 - 6q3/2 + 5q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-14 + 2q-10 + 4q-8 + q-6 + 3q-4 - 2q4 + q6 - q8 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 + a-1z-1 - 2a-1z3 - a-1z5 - 3az-1 - 5az - 3az3 - az5 + 2a3z-1 + 2a3z + a3z3

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + a-3z - 7a-3z3 + 10a-3z5 - 3a-3z7 - a-2 + a-2z2 - 5a-2z4 + 9a-2z6 - 3a-2z8 + a-1z-1 - a-1z - 5a-1z3 + 9a-1z5 - a-1z7 - a-1z9 - 3 + 9z2 - 13z4 + 12z6 - 4z8 + 3az-1 - 9az + 9az3 - 5az5 + 2az7 - az9 - 3a2 + 7a2z2 - 7a2z4 + 2a2z6 - a2z8 + 2a3z-1 - 6a3z + 6a3z3 - 4a3z5 + a4z2 - 2a4z4 + a5z - a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               3 1   j = 4             3 2     j = 2           4 3       j = 0         4 3         j = -2       3 5           j = -4     2 3             j = -6     3               j = -8 1 2                 j = -10 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 56]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 56]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 17, 12, 16], X[14, 7, 15, 8], > X[8, 15, 9, 16], X[13, 21, 14, 20], X[17, 5, 18, 22], X[21, 19, 22, 18], > X[19, 13, 20, 12], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, -3, 9, -6, -4, 5, 3, -7, 8, > -9, 6, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 2 5 6 8 3/2 5/2 -q + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 6 q + 5 q - 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 4 -6 3 4 6 8 12 14 q + --- + -- + q + -- - 2 q + q - q + q - q 10 8 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 56]][a, z]
Out[8]=	3 3 3 5 1 3 a 2 a z 3 z 2 z 3 3 3 z 5 --- - --- + ---- + -- - 5 a z + 2 a z + -- - ---- - 3 a z + a z - -- - a z a z z z 3 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 56]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a z z 3 5 2 -3 - a - 3 a + --- + --- + ---- + -- - - - 9 a z - 6 a z + a z + 9 z - a z z z 3 a a 2 2 3 3 2 z z 2 2 4 2 7 z 5 z 3 3 3 5 3 > ---- + -- + 7 a z + a z - ---- - ---- + 9 a z + 6 a z - a z - 4 2 3 a a a a 4 4 5 5 4 3 z 5 z 2 4 4 4 10 z 9 z 5 3 5 > 13 z + ---- - ---- - 7 a z - 2 a z + ----- + ---- - 5 a z - 4 a z + 4 2 3 a a a a 6 6 7 7 8 6 z 9 z 2 6 3 z z 7 8 3 z 2 8 > 12 z - -- + ---- + 2 a z - ---- - -- + 2 a z - 4 z - ---- - a z - 4 2 3 a 2 a a a a 9 z 9 > -- - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 1 2 3 2 3 3 2 4 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 4 q t + 2 10 4 8 4 8 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n56

L11n55 L11n57