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L11a347 L11a349

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DrawMorseLink

PD Presentation:

X12,1,13,2 X10,11,1,12 X14,5,15,6 X18,9,19,10 X2,17,3,18 X16,8,17,7 X8493 X20,16,21,15 X22,13,11,14 X4,20,5,19 X6,21,7,22

Gauss Code:

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

Jones Polynomial:

q-15/2 - 6q-13/2 + 13q-11/2 - 20q-9/2 + 26q-7/2 - 30q-5/2 + 28q-3/2 - 25q-1/2 + 17q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 4q-20 - 2q-18 + q-16 + 3q-14 - 6q-12 + 6q-10 - q-8 + 4q-6 + 4q-4 - 4q-2 + 5 - 5q2 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 + 4az + 5az3 + 3az5 + az7 - a3z-1 - 8a3z - 5a3z3 + a3z5 + a3z7 + a5z-1 + 3a5z - a5z5

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 4a-2z6 + 2a-1z - 6a-1z3 + 10a-1z5 - 8a-1z7 - 2z2 - 5z4 + 13z6 - 11z8 + 7az - 18az3 + 11az5 + 6az7 - 10az9 + 3a2z2 - 33a2z4 + 49a2z6 - 18a2z8 - 4a2z10 - a3z-1 + 10a3z - 17a3z3 - 2a3z5 + 36a3z7 - 22a3z9 + a4 + a4z2 - 33a4z4 + 59a4z6 - 20a4z8 - 4a4z10 - a5z-1 + 3a5z - 6a5z3 + 6a5z5 + 16a5z7 - 12a5z9 - 2a6z2 - 10a6z4 + 26a6z6 - 13a6z8 - 2a7z + 8a7z5 - 6a7z7 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 11 3     j = 0               14 6       j = -2             15 12         j = -4           15 13           j = -6         11 15             j = -8       9 15               j = -10     5 12                 j = -12   1 8                   j = -14   5                     j = -16 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, Alternating, 348]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 348]]
Out[4]=	PD[X[12, 1, 13, 2], X[10, 11, 1, 12], X[14, 5, 15, 6], X[18, 9, 19, 10], > X[2, 17, 3, 18], X[16, 8, 17, 7], X[8, 4, 9, 3], X[20, 16, 21, 15], > X[22, 13, 11, 14], X[4, 20, 5, 19], X[6, 21, 7, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 7, -10, 3, -11, 6, -7, 4, -2}, > {2, -1, 9, -3, 8, -6, 5, -4, 10, -8, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 6 13 20 26 30 28 25 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 17 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 4 2 -16 3 6 6 -8 4 4 4 2 5 - q + --- - --- + q + --- - --- + --- - q + -- + -- - -- - 5 q + 20 18 14 12 10 6 4 2 q q q q q q q q 6 8 10 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 348]][a, z]
Out[8]=	3 5 3 5 a a z 3 5 2 z 3 3 3 z -(--) + -- - - + 4 a z - 8 a z + 3 a z - ---- + 5 a z - 5 a z - -- + z z a a a 5 3 5 5 5 7 3 7 > 3 a z + a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 348]][a, z]
Out[9]=	3 5 2 4 a a 2 z 3 5 7 2 2 z a - -- - -- + --- + 7 a z + 10 a z + 3 a z - 2 a z - 2 z - ---- + z z a 2 a 3 3 2 2 4 2 6 2 z 6 z 3 3 3 5 3 > 3 a z + a z - 2 a z + -- - ---- - 18 a z - 17 a z - 6 a z - 3 a a 4 5 5 4 5 z 2 4 4 4 6 4 z 10 z 5 > 5 z + ---- - 33 a z - 33 a z - 10 a z - -- + ----- + 11 a z - 2 3 a a a 6 3 5 5 5 7 5 6 4 z 2 6 4 6 > 2 a z + 6 a z + 8 a z + 13 z - ---- + 49 a z + 59 a z + 2 a 7 6 6 8 6 8 z 7 3 7 5 7 7 7 8 > 26 a z - a z - ---- + 6 a z + 36 a z + 16 a z - 6 a z - 11 z - a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 18 a z - 20 a z - 13 a z - 10 a z - 22 a z - 12 a z - 4 a z - 4 10 > 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 5 1 8 5 12 9 15 14 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 11 15 15 13 15 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 11 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a348

L11a347 L11a349