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L11a496 L11a498

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,8,15,7 X20,11,21,12 X18,10,5,9 X10,19,11,20 X8,16,9,15 X16,21,17,22 X22,17,19,18 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 8q-4 - 11q-3 + 16q-2 - 17q-1 + 19 - 15q + 11q2 - 7q3 + 3q4 - q5

A2 (sl(3)) Invariant:

q-18 + 2q-14 + 5q-12 + 2q-10 + 8q-8 + 4q-6 + 4q-4 + 6q-2 - 1 + 4q2 - 4q4 - q6 - 3q10 + q12 - q14

HOMFLY-PT Polynomial:

- a-2z-2 - 4a-2 - 6a-2z2 - 4a-2z4 - a-2z6 + 4z-2 + 14 + 19z2 + 15z4 + 6z6 + z8 - 5a2z-2 - 14a2 - 16a2z2 - 9a2z4 - 2a2z6 + 2a4z-2 + 4a4 + 3a4z2 + a4z4

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 + a-4z2 - 5a-4z4 + 3a-4z6 + a-3z-1 - 4a-3z + 4a-3z3 - 8a-3z5 + 5a-3z7 - a-2z-2 + 5a-2 - 11a-2z2 + 12a-2z4 - 11a-2z6 + 6a-2z8 + 5a-1z-1 - 23a-1z + 37a-1z3 - 21a-1z5 + a-1z7 + 4a-1z9 - 4z-2 + 19 - 46z2 + 72z4 - 50z6 + 14z8 + z10 + 9az-1 - 35az + 50az3 - 19az5 - 10az7 + 8az9 - 5a2z-2 + 22a2 - 50a2z2 + 68a2z4 - 49a2z6 + 13a2z8 + a2z10 + 5a3z-1 - 17a3z + 22a3z3 - 14a3z5 - 3a3z7 + 4a3z9 - 2a4z-2 + 8a4 - 13a4z2 + 10a4z4 - 12a4z6 + 5a4z8 + 3a5z3 - 7a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     2   j = 7                   5 1   j = 5                 6 2     j = 3               9 5       j = 1             10 6         j = -1           9 11           j = -3         7 8             j = -5       4 9               j = -7     4 7                 j = -9   1 6                   j = -11   2                     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, Alternating, 497]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 497]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[20, 11, 21, 12], > X[18, 10, 5, 9], X[10, 19, 11, 20], X[8, 16, 9, 15], X[16, 21, 17, 22], > X[22, 17, 19, 18], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {6, -4, 8, -9}, > {10, -1, 3, -7, 5, -6, 4, -2, 11, -3, 7, -8, 9, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 8 11 16 17 2 3 4 5 19 + q - -- + -- - -- + -- - -- - 15 q + 11 q - 7 q + 3 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 5 2 8 4 4 6 2 4 6 10 -1 + q + --- + --- + --- + -- + -- + -- + -- + 4 q - 4 q - q - 3 q + 14 12 10 8 6 4 2 q q q q q q q 12 14 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 497]][a, z]
Out[8]=	2 4 2 4 2 4 4 1 5 a 2 a 2 6 z 2 2 14 - -- - 14 a + 4 a + -- - ----- - ---- + ---- + 19 z - ---- - 16 a z + 2 2 2 2 2 2 2 a z a z z z a 4 6 4 2 4 4 z 2 4 4 4 6 z 2 6 8 > 3 a z + 15 z - ---- - 9 a z + a z + 6 z - -- - 2 a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 497]][a, z]
Out[9]=	2 4 5 2 4 6 4 1 5 a 2 a 1 5 9 a 19 + -- + 22 a + 8 a - a - -- - ----- - ---- - ---- + ---- + --- + --- + 2 2 2 2 2 2 3 a z z a z a z z z a z 3 2 2 5 a z 4 z 23 z 3 2 z 11 z 2 2 > ---- + -- - --- - ---- - 35 a z - 17 a z - 46 z + -- - ----- - 50 a z - z 5 3 a 4 2 a a a a 3 3 3 4 2 6 2 2 z 4 z 37 z 3 3 3 5 3 > 13 a z + 3 a z - ---- + ---- + ----- + 50 a z + 22 a z + 3 a z + 5 3 a a a 4 4 5 5 5 4 5 z 12 z 2 4 4 4 6 4 z 8 z 21 z > 72 z - ---- + ----- + 68 a z + 10 a z - 3 a z + -- - ---- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 11 z 2 6 4 6 > 19 a z - 14 a z - 7 a z - 50 z + ---- - ----- - 49 a z - 12 a z + 4 2 a a 7 7 8 6 6 5 z z 7 3 7 5 7 8 6 z 2 8 > a z + ---- + -- - 10 a z - 3 a z + 3 a z + 14 z + ---- + 13 a z + 3 a 2 a a 9 4 8 4 z 9 3 9 10 2 10 > 5 a z + ---- + 8 a z + 4 a z + z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 2 1 6 4 7 4 9 -- + 10 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 7 8 9 3 3 2 5 2 5 3 > ----- + ---- + --- + 6 q t + 9 q t + 5 q t + 6 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a497

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