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L11n314 L11n316

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,17,8,16 X9,20,10,21 X11,18,12,19 X19,22,20,11 X15,9,16,8 X21,10,22,5 X17,14,18,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-22 + q-14 + 2q-10 + q-8 + 2q-6 + 3q-4 + 3q-2 + 5 + 4q2 + 3q4 + 2q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - 2z-2 - 4 - 4z2 - z4 + a2z-2 + a2 + 2a4 + 3a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial:

a-2z-2 - 4a-2 + 7a-2z2 - 5a-2z4 + a-2z6 - 2a-1z-1 + 4a-1z + a-1z3 - 4a-1z5 + a-1z7 + 2z-2 - 7 + 14z2 - 11z4 + 2z6 - 2az-1 + 6az - 5az3 + a2z-2 - 2a2 - 5a2z2 + 11a2z4 - 7a2z6 + a2z8 - 3a3z3 + 8a3z5 - 6a3z7 + a3z9 + 4a4 - 20a4z2 + 33a4z4 - 19a4z6 + 3a4z8 - 4a5z + 9a5z3 - a5z5 - 4a5z7 + a5z9 + 2a6 - 8a6z2 + 16a6z4 - 11a6z6 + 2a6z8 - 2a7z + 6a7z3 - 5a7z5 + a7z7

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 = 7                       1 j = 5                         j = 3                   2 1   j = 1               2 1 1     j = -1             1 3 1       j = -3           2 2 2         j = -5         2 3 1           j = -7       1 1 1             j = -9     1 2 1               j = -11   1 1                   j = -13   1                     j = -15 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, 315]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 315]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 17, 8, 16], X[9, 20, 10, 21], > X[11, 18, 12, 19], X[19, 22, 20, 11], X[15, 9, 16, 8], X[21, 10, 22, 5], > X[17, 14, 18, 15], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 7, -4, 8}, > {-5, -2, 11, 9, -7, 3, -9, 5, -6, 4, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 2 2 -3 -2 1 2 3 -q + -- - -- + -- - q + q + - + 2 q - q + q 6 5 4 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -14 2 -8 2 3 3 2 4 6 8 10 5 - q + q + --- + q + -- + -- + -- + 4 q + 3 q + 2 q + q + q 10 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 315]][a, z]
Out[8]=	2 2 2 2 4 6 2 1 a 2 z 4 2 6 2 4 -4 + -- + a + 2 a - a - -- + ----- + -- - 4 z + -- + 3 a z - a z - z + 2 2 2 2 2 2 a z a z z a 4 4 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 315]][a, z]
Out[9]=	2 4 2 4 6 2 1 a 2 2 a 4 z -7 - -- - 2 a + 4 a + 2 a + -- + ----- + -- - --- - --- + --- + 6 a z - 2 2 2 2 2 a z z a a z a z z 2 3 5 7 2 7 z 2 2 4 2 6 2 z > 4 a z - 2 a z + 14 z + ---- - 5 a z - 20 a z - 8 a z + -- - 2 a a 4 3 3 3 5 3 7 3 4 5 z 2 4 4 4 > 5 a z - 3 a z + 9 a z + 6 a z - 11 z - ---- + 11 a z + 33 a z + 2 a 5 6 6 4 4 z 3 5 5 5 7 5 6 z 2 6 > 16 a z - ---- + 8 a z - a z - 5 a z + 2 z + -- - 7 a z - a 2 a 7 4 6 6 6 z 3 7 5 7 7 7 2 8 4 8 > 19 a z - 11 a z + -- - 6 a z - 4 a z + a z + a z + 3 a z + a 6 8 3 9 5 9 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 1 1 2 1 -- + - + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q q t q t q t q t q t q t q t 1 1 2 1 3 2 1 2 1 t > ----- + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + - + 9 3 7 3 5 3 7 2 5 2 3 2 5 3 q t q q t q t q t q t q t q t q t q t 2 3 2 3 3 7 4 > q t + q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n315

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