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L11n252 L11n254

Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X14,3,15,4 X9,18,10,19 X5,16,6,17 X22,7,11,8 X6,21,7,22 X20,15,21,16 X17,8,18,9 X19,4,20,5 X2,11,3,12 X10,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-25/2 + q-17/2 - q-15/2 + q-13/2 - 2q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 + q-40 + 2q-38 + q-36 + 2q-34 - q-28 + q-24 + q-20 + q-18 + q-16 + q-12

HOMFLY-PT Polynomial:

- 5a7z - 10a7z3 - 6a7z5 - a7z7 - a9z-1 - 6a9z - 10a9z3 - 6a9z5 - a9z7 + a11z-1 + 7a11z + 6a11z3 + a11z5 - a13z

Kauffman Polynomial:

5a7z - 10a7z3 + 6a7z5 - a7z7 - 5a8z4 + 5a8z6 - a8z8 + a9z-1 - 7a9z + 11a9z3 - 11a9z5 + 6a9z7 - a9z9 - a10 + 9a10z2 - 19a10z4 + 12a10z6 - 2a10z8 + a11z-1 - 9a11z + 18a11z3 - 16a11z5 + 7a11z7 - a11z9 + 10a12z2 - 14a12z4 + 7a12z6 - a12z8 - 2a13z + 2a13z3 + a14z2 - 5a15z + 5a15z3 - a15z5

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                     1 j = -8                   1 1 j = -10                 1     j = -12             1 1 1     j = -14           1 2 1       j = -16           1 1         j = -18       1 2 2           j = -20                       j = -22     1 1               j = -24 1                     j = -26 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, 253]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 253]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 3, 15, 4], X[9, 18, 10, 19], X[5, 16, 6, 17], > X[22, 7, 11, 8], X[6, 21, 7, 22], X[20, 15, 21, 16], X[17, 8, 18, 9], > X[19, 4, 20, 5], X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, 9, -4, -6, 5, 8, -3, -11}, > {10, -1, 11, -2, 7, 4, -8, 3, -9, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) -(17/2) -(15/2) -(13/2) 2 -(9/2) -(7/2) -q + q - q + q - ----- + q - q 11/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 2 -36 2 -28 -24 -20 -18 -16 -12 -q + q + --- + q + --- - q + q + q + q + q + q 38 34 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 253]][a, z]
Out[8]=	9 11 a a 7 9 11 13 7 3 9 3 -(--) + --- - 5 a z - 6 a z + 7 a z - a z - 10 a z - 10 a z + z z 11 3 7 5 9 5 11 5 7 7 9 7 > 6 a z - 6 a z - 6 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 253]][a, z]
Out[9]=	9 11 10 a a 7 9 11 13 15 10 2 -a + -- + --- + 5 a z - 7 a z - 9 a z - 2 a z - 5 a z + 9 a z + z z 12 2 14 2 7 3 9 3 11 3 13 3 > 10 a z + a z - 10 a z + 11 a z + 18 a z + 2 a z + 15 3 8 4 10 4 12 4 7 5 9 5 > 5 a z - 5 a z - 19 a z - 14 a z + 6 a z - 11 a z - 11 5 15 5 8 6 10 6 12 6 7 7 9 7 > 16 a z - a z + 5 a z + 12 a z + 7 a z - a z + 6 a z + 11 7 8 8 10 8 12 8 9 9 11 9 > 7 a z - a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 1 1 2 2 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 26 10 24 10 22 8 22 7 18 7 18 6 18 5 q t q t q t q t q t q t q t 1 1 1 2 1 1 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 16 5 14 5 16 4 14 4 12 4 14 3 12 3 12 2 q t q t q t q t q t q t q t q t 1 1 > ------ + ---- 10 2 8 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n253

L11n252 L11n254