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L11a356 L11a358

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DrawMorseLink

PD Presentation:

X12,1,13,2 X18,7,19,8 X14,3,15,4 X20,10,21,9 X22,19,11,20 X6,15,7,16 X16,5,17,6 X4,17,5,18 X8,22,9,21 X2,11,3,12 X10,13,1,14

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 4q-17/2 - 9q-15/2 + 11q-13/2 - 14q-11/2 + 15q-9/2 - 13q-7/2 + 10q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 - q-26 + 4q-24 + q-22 + q-20 + 4q-18 - q-16 + 2q-14 - 2q-12 - q-10 + 2q-8 - 2q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - 3a3z + a3z5 - a5z-1 + a5z + 4a5z3 + 2a5z5 + a7z-1 + 2a7z + 2a7z3 + a7z5 - 2a9z - a9z3

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 5a2z4 - 3a2z6 + 3a3z - 5a3z3 + 9a3z5 - 5a3z7 + 2a4z2 - 5a4z4 + 8a4z6 - 5a4z8 + a5z-1 - 2a5z - 2a5z3 + 3a5z5 + a5z7 - 3a5z9 - a6 - a6z2 - 5a6z4 + 8a6z6 - 4a6z8 - a6z10 + a7z-1 - 4a7z + 11a7z3 - 15a7z5 + 11a7z7 - 5a7z9 - 4a8z2 + 4a8z4 + a8z6 - a8z8 - a8z10 + 6a9z - 2a9z3 - a9z5 + 3a9z7 - 2a9z9 - 4a10z2 + 3a10z4 + 3a10z6 - 2a10z8 + 4a11z - 8a11z3 + 7a11z5 - 2a11z7 - 4a12z2 + 4a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     2   j = -2                   5 1   j = -4                 6 3     j = -6               7 4       j = -8             8 6         j = -10           6 7           j = -12         5 8             j = -14       4 6               j = -16     1 6                 j = -18   1 3                   j = -20   1                     j = -22 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, 357]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 357]]
Out[4]=	PD[X[12, 1, 13, 2], X[18, 7, 19, 8], X[14, 3, 15, 4], X[20, 10, 21, 9], > X[22, 19, 11, 20], X[6, 15, 7, 16], X[16, 5, 17, 6], X[4, 17, 5, 18], > X[8, 22, 9, 21], X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 3, -8, 7, -6, 2, -9, 4, -11}, > {10, -1, 11, -3, 6, -7, 8, -2, 5, -4, 9, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 4 9 11 14 15 13 10 7 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -26 4 -22 -20 4 -16 2 2 -10 2 -1 - q - q + --- + q + q + --- - q + --- - --- - q + -- - 24 18 14 12 8 q q q q q 2 3 2 > -- + -- + q 6 4 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 357]][a, z]
Out[8]=	5 7 a a 3 5 7 9 3 5 3 7 3 -(--) + -- - a z - 3 a z + a z + 2 a z - 2 a z - a z + 4 a z + 2 a z - z z 9 3 3 5 5 5 7 5 > a z + a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 357]][a, z]
Out[9]=	5 7 6 a a 3 5 7 9 11 2 2 -a + -- + -- - a z + 3 a z - 2 a z - 4 a z + 6 a z + 4 a z - a z + z z 4 2 6 2 8 2 10 2 12 2 3 3 3 > 2 a z - a z - 4 a z - 4 a z - 4 a z + 2 a z - 5 a z - 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 2 a z + 11 a z - 2 a z - 8 a z + 5 a z - 5 a z - 5 a z + 8 4 10 4 12 4 5 3 5 5 5 7 5 > 4 a z + 3 a z + 4 a z - a z + 9 a z + 3 a z - 15 a z - 9 5 11 5 2 6 4 6 6 6 8 6 10 6 > a z + 7 a z - 3 a z + 8 a z + 8 a z + a z + 3 a z - 12 6 3 7 5 7 7 7 9 7 11 7 4 8 > a z - 5 a z + a z + 11 a z + 3 a z - 2 a z - 5 a z - 6 8 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 4 a z - a z - 2 a z - 3 a z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 1 1 3 1 6 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 6 5 8 6 7 8 6 7 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 6 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a357

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