© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a350 K11a352

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X20,9,21,10 X4,12,5,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X8,19,9,20 X10,21,11,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 18 16 14 20 4 2 22 12 8 10

Alexander Polynomial:

t-4 - 7t-3 + 20t-2 - 34t-1 + 41 - 34t + 20t2 - 7t3 + t4

Conway Polynomial:

1 - z2 - 2z4 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{165, 0}

Jones Polynomial:

- q-5 + 5q-4 - 11q-3 + 18q-2 - 24q-1 + 27 - 26q + 23q2 - 16q3 + 9q4 - 4q5 + q6

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-14 + 3q-12 - 3q-10 + 3q-8 + q-6 - 4q-4 + 5q-2 - 6 + 4q2 - q4 + 5q8 - 4q10 + 2q12 - q14 - q16 + q18

HOMFLY-PT Polynomial:

2a-4z2 + a-4z4 + a-2 - 4a-2z2 - 6a-2z4 - 2a-2z6 - 1 + z2 + 5z4 + 4z6 + z8 + a2 - 2a2z4 - a2z6

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 4a-5z3 - 9a-5z5 + 4a-5z7 - 6a-4z2 + 17a-4z4 - 20a-4z6 + 8a-4z8 + a-3z - 5a-3z3 + 16a-3z5 - 20a-3z7 + 9a-3z9 - a-2 - 11a-2z2 + 38a-2z4 - 37a-2z6 + 8a-2z8 + 4a-2z10 + 2a-1z - 13a-1z3 + 37a-1z5 - 45a-1z7 + 20a-1z9 - 1 - 4z2 + 27z4 - 40z6 + 14z8 + 4z10 + 2az - az3 - 3az5 - 10az7 + 11az9 - a2 + a2z2 + 4a2z4 - 19a2z6 + 14a2z8 + a3z + 3a3z3 - 14a3z5 + 11a3z7 - 4a4z4 + 5a4z6 + a5z5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-1, 0}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 11351. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   6 1   j = 7                 10 3     j = 5               13 6       j = 3             13 10         j = 1           14 13           j = -1         11 14             j = -3       7 13               j = -5     4 11                 j = -7   1 7                   j = -9   4                     j = -11 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 351]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 351]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 4, 19, 3], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[20, 9, 21, 10], X[4, 12, 5, 11], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[12, 18, 13, 17], X[8, 19, 9, 20], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 351]]
Out[4]=	GaussCode[1, -7, 2, -6, 3, -1, 4, -10, 5, -11, 6, -9, 7, -4, 8, -3, 9, -2, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 351]]
Out[5]=	DTCode[6, 18, 16, 14, 20, 4, 2, 22, 12, 8, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 351]][t]
Out[6]=	-4 7 20 34 2 3 4 41 + t - -- + -- - -- - 34 t + 20 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 351]][z]
Out[7]=	2 4 6 8 1 - z - 2 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 351]}
In[9]:=	{KnotDet[Knot[11, Alternating, 351]], KnotSignature[Knot[11, Alternating, 351]]}
Out[9]=	{165, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 351]][q]
Out[10]=	-5 5 11 18 24 2 3 4 5 6 27 - q + -- - -- + -- - -- - 26 q + 23 q - 16 q + 9 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 351]}
In[12]:=	A2Invariant[Knot[11, Alternating, 351]][q]
Out[12]=	-14 3 3 3 -6 4 5 2 4 8 10 12 -6 - q + --- - --- + -- + q - -- + -- + 4 q - q + 5 q - 4 q + 2 q - 12 10 8 4 2 q q q q q 14 16 18 > q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 351]][a, z]
Out[13]=	2 2 4 4 6 -2 2 2 2 z 4 z 4 z 6 z 2 4 6 2 z -1 + a + a + z + ---- - ---- + 5 z + -- - ---- - 2 a z + 4 z - ---- - 4 2 4 2 2 a a a a a 2 6 8 > a z + z
In[14]:=	Kauffman[Knot[11, Alternating, 351]][a, z]
Out[14]=	2 2 3 -2 2 z 2 z 3 2 6 z 11 z 2 2 4 z -1 - a - a + -- + --- + 2 a z + a z - 4 z - ---- - ----- + a z + ---- - 3 a 4 2 5 a a a a 3 3 4 4 4 5 z 13 z 3 3 3 4 2 z 17 z 38 z 2 4 > ---- - ----- - a z + 3 a z + 27 z - ---- + ----- + ----- + 4 a z - 3 a 6 4 2 a a a a 5 5 5 6 4 4 9 z 16 z 37 z 5 3 5 5 5 6 z > 4 a z - ---- + ----- + ----- - 3 a z - 14 a z + a z - 40 z + -- - 5 3 a 6 a a a 6 6 7 7 7 20 z 37 z 2 6 4 6 4 z 20 z 45 z 7 > ----- - ----- - 19 a z + 5 a z + ---- - ----- - ----- - 10 a z + 4 2 5 3 a a a a a 8 8 9 9 3 7 8 8 z 8 z 2 8 9 z 20 z 9 > 11 a z + 14 z + ---- + ---- + 14 a z + ---- + ----- + 11 a z + 4 2 3 a a a a 10 10 4 z > 4 z + ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 351]], Vassiliev[3][Knot[11, Alternating, 351]]}
Out[15]=	{-1, 0}
In[16]:=	Kh[Knot[11, Alternating, 351]][q, t]
Out[16]=	14 1 4 1 7 4 11 7 13 -- + 14 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 11 3 3 2 5 2 5 3 7 3 > --- + 13 q t + 13 q t + 10 q t + 13 q t + 6 q t + 10 q t + q t 7 4 9 4 9 5 11 5 13 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a351

K11a350 K11a352