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

K11a349 K11a351

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 18 16 14 4 2 20 22 10 12 8

Alexander Polynomial:

t-4 - 7t-3 + 21t-2 - 39t-1 + 49 - 39t + 21t2 - 7t3 + t4

Conway Polynomial:

1 - 2z2 - z4 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{185, 0}

Jones Polynomial:

- q-5 + 5q-4 - 11q-3 + 19q-2 - 26q-1 + 30 - 30q + 26q2 - 19q3 + 12q4 - 5q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + 3q-12 - 3q-10 + 4q-8 + 2q-6 - 5q-4 + 5q-2 - 7 + 3q2 - q4 - q6 + 6q8 - 4q10 + 3q12 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 + 2a-2 - 2a-2z2 - 5a-2z4 - 2a-2z6 - 3 - z2 + 5z4 + 4z6 + z8 + 2a2 - 2a2z4 - a2z6

Kauffman Polynomial:

- a-6z4 + a-6z6 + 2a-5z3 - 8a-5z5 + 5a-5z7 - 3a-4z2 + 13a-4z4 - 23a-4z6 + 11a-4z8 - a-3z3 + 9a-3z5 - 22a-3z7 + 12a-3z9 - 2a-2 - 3a-2z2 + 35a-2z4 - 49a-2z6 + 13a-2z8 + 5a-2z10 - 2a-1z + a-1z3 + 27a-1z5 - 51a-1z7 + 25a-1z9 - 3 + 3z2 + 30z4 - 50z6 + 17z8 + 5z10 - 2az + 7az3 - 4az5 - 13az7 + 13az9 - 2a2 + 3a2z2 + 5a2z4 - 20a2z6 + 15a2z8 + 3a3z3 - 13a3z5 + 11a3z7 - 4a4z4 + 5a4z6 + a5z5

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

{-2, 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 11350. 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                     4   j = 9                   8 1   j = 7                 11 4     j = 5               15 8       j = 3             15 11         j = 1           15 15           j = -1         12 16             j = -3       7 14               j = -5     4 12                 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, 350]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 350]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 4, 19, 3], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[4, 10, 5, 9], X[2, 11, 3, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[10, 18, 11, 17], X[12, 19, 13, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 350]]
Out[4]=	GaussCode[1, -6, 2, -5, 3, -1, 4, -11, 5, -9, 6, -10, 7, -4, 8, -3, 9, -2, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 350]]
Out[5]=	DTCode[6, 18, 16, 14, 4, 2, 20, 22, 10, 12, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 350]][t]
Out[6]=	-4 7 21 39 2 3 4 49 + t - -- + -- - -- - 39 t + 21 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 350]][z]
Out[7]=	2 4 6 8 1 - 2 z - z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 350]}
In[9]:=	{KnotDet[Knot[11, Alternating, 350]], KnotSignature[Knot[11, Alternating, 350]]}
Out[9]=	{185, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 350]][q]
Out[10]=	-5 5 11 19 26 2 3 4 5 6 30 - q + -- - -- + -- - -- - 30 q + 26 q - 19 q + 12 q - 5 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, 350]}
In[12]:=	A2Invariant[Knot[11, Alternating, 350]][q]
Out[12]=	-14 3 3 4 2 5 5 2 4 6 8 10 -7 - q + --- - --- + -- + -- - -- + -- + 3 q - q - q + 6 q - 4 q + 12 10 8 6 4 2 q q q q q q 12 16 18 > 3 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 350]][a, z]
Out[13]=	2 2 4 4 6 2 2 2 z 2 z 4 z 5 z 2 4 6 2 z -3 + -- + 2 a - z + -- - ---- + 5 z + -- - ---- - 2 a z + 4 z - ---- - 2 4 2 4 2 2 a a a a a a 2 6 8 > a z + z
In[14]:=	Kauffman[Knot[11, Alternating, 350]][a, z]
Out[14]=	2 2 3 3 3 2 2 2 z 2 3 z 3 z 2 2 2 z z z -3 - -- - 2 a - --- - 2 a z + 3 z - ---- - ---- + 3 a z + ---- - -- + -- + 2 a 4 2 5 3 a a a a a a 4 4 4 5 3 3 3 4 z 13 z 35 z 2 4 4 4 8 z > 7 a z + 3 a z + 30 z - -- + ----- + ----- + 5 a z - 4 a z - ---- + 6 4 2 5 a a a a 5 5 6 6 6 9 z 27 z 5 3 5 5 5 6 z 23 z 49 z > ---- + ----- - 4 a z - 13 a z + a z - 50 z + -- - ----- - ----- - 3 a 6 4 2 a a a a 7 7 7 2 6 4 6 5 z 22 z 51 z 7 3 7 8 > 20 a z + 5 a z + ---- - ----- - ----- - 13 a z + 11 a z + 17 z + 5 3 a a a 8 8 9 9 10 11 z 13 z 2 8 12 z 25 z 9 10 5 z > ----- + ----- + 15 a z + ----- + ----- + 13 a z + 5 z + ----- 4 2 3 a 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 350]], Vassiliev[3][Knot[11, Alternating, 350]]}
Out[15]=	{-2, 0}
In[16]:=	Kh[Knot[11, Alternating, 350]][q, t]
Out[16]=	16 1 4 1 7 4 12 7 14 -- + 15 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 12 3 3 2 5 2 5 3 7 3 > --- + 15 q t + 15 q t + 11 q t + 15 q t + 8 q t + 11 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a350

K11a349 K11a351