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

K11a249 K11a251

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8493 X12,5,13,6 X2837 X14,10,15,9 X18,11,19,12 X4,13,5,14 X20,16,21,15 X22,18,1,17 X10,19,11,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 12 2 14 18 4 20 22 10 16

Alexander Polynomial:

t-4 - 5t-3 + 11t-2 - 16t-1 + 19 - 16t + 11t2 - 5t3 + t4

Conway Polynomial:

1 - z2 + z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{85, 4}

Jones Polynomial:

q-2 - 2q-1 + 5 - 8q + 11q2 - 13q3 + 13q4 - 12q5 + 10q6 - 6q7 + 3q8 - q9

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

{...}

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + 2 - 2q2 - 2q6 - 2q8 + 2q10 - 2q12 + 4q14 + q18 + q20 - 2q22 + q24 - q26

HOMFLY-PT Polynomial:

- 2a-6 - 5a-6z2 - 4a-6z4 - a-6z6 + 7a-4 + 16a-4z2 + 14a-4z4 + 6a-4z6 + a-4z8 - 8a-2 - 16a-2z2 - 10a-2z4 - 2a-2z6 + 4 + 4z2 + z4

Kauffman Polynomial:

a-11z3 + 3a-10z4 - 3a-9z3 + 6a-9z5 + 6a-8z2 - 15a-8z4 + 10a-8z6 - 4a-7z + 15a-7z3 - 26a-7z5 + 12a-7z7 + 2a-6 - 2a-6z2 + 9a-6z4 - 23a-6z6 + 10a-6z8 - 9a-5z + 31a-5z3 - 26a-5z5 - 4a-5z7 + 5a-5z9 + 7a-4 - 33a-4z2 + 68a-4z4 - 54a-4z6 + 11a-4z8 + a-4z10 - 5a-3z + 4a-3z3 + 22a-3z5 - 26a-3z7 + 7a-3z9 + 8a-2 - 37a-2z2 + 54a-2z4 - 27a-2z6 + 2a-2z8 + a-2z10 - 8a-1z3 + 16a-1z5 - 10a-1z7 + 2a-1z9 + 4 - 12z2 + 13z4 - 6z6 + z8

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

{-1, 1}

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=4 is the signature of 11250. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                       1 j = 17                     2   j = 15                   4 1   j = 13                 6 2     j = 11               6 4       j = 9             7 6         j = 7           6 6           j = 5         5 7             j = 3       4 7               j = 1     1 4                 j = -1   1 4                   j = -3   1                     j = -5 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, 250]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 250]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[14, 10, 15, 9], X[18, 11, 19, 12], X[4, 13, 5, 14], X[20, 16, 21, 15], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 250]]
Out[4]=	GaussCode[1, -4, 2, -7, 3, -1, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 250]]
Out[5]=	DTCode[6, 8, 12, 2, 14, 18, 4, 20, 22, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 250]][t]
Out[6]=	-4 5 11 16 2 3 4 19 + t - -- + -- - -- - 16 t + 11 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 250]][z]
Out[7]=	2 4 6 8 1 - z + z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 250]}
In[9]:=	{KnotDet[Knot[11, Alternating, 250]], KnotSignature[Knot[11, Alternating, 250]]}
Out[9]=	{85, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 250]][q]
Out[10]=	-2 2 2 3 4 5 6 7 8 9 5 + q - - - 8 q + 11 q - 13 q + 13 q - 12 q + 10 q - 6 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 250]}
In[12]:=	A2Invariant[Knot[11, Alternating, 250]][q]
Out[12]=	-6 -4 -2 2 6 8 10 12 14 18 20 2 + q + q + q - 2 q - 2 q - 2 q + 2 q - 2 q + 4 q + q + q - 22 24 26 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 250]][a, z]
Out[13]=	2 2 2 4 4 4 2 7 8 2 5 z 16 z 16 z 4 4 z 14 z 10 z 4 - -- + -- - -- + 4 z - ---- + ----- - ----- + z - ---- + ----- - ----- - 6 4 2 6 4 2 6 4 2 a a a a a a a a a 6 6 6 8 z 6 z 2 z z > -- + ---- - ---- + -- 6 4 2 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 250]][a, z]
Out[14]=	2 2 2 2 2 7 8 4 z 9 z 5 z 2 6 z 2 z 33 z 37 z 4 + -- + -- + -- - --- - --- - --- - 12 z + ---- - ---- - ----- - ----- + 6 4 2 7 5 3 8 6 4 2 a a a a a a a a a a 3 3 3 3 3 3 4 4 4 z 3 z 15 z 31 z 4 z 8 z 4 3 z 15 z 9 z > --- - ---- + ----- + ----- + ---- - ---- + 13 z + ---- - ----- + ---- + 11 9 7 5 3 a 10 8 6 a a a a a a a a 4 4 5 5 5 5 5 6 68 z 54 z 6 z 26 z 26 z 22 z 16 z 6 10 z > ----- + ----- + ---- - ----- - ----- + ----- + ----- - 6 z + ----- - 4 2 9 7 5 3 a 8 a a a a a a a 6 6 6 7 7 7 7 8 8 23 z 54 z 27 z 12 z 4 z 26 z 10 z 8 10 z 11 z > ----- - ----- - ----- + ----- - ---- - ----- - ----- + z + ----- + ----- + 6 4 2 7 5 3 a 6 4 a a a a a a a a 8 9 9 9 10 10 2 z 5 z 7 z 2 z z z > ---- + ---- + ---- + ---- + --- + --- 2 5 3 a 4 2 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 250]], Vassiliev[3][Knot[11, Alternating, 250]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, Alternating, 250]][q, t]
Out[16]=	3 3 5 1 1 1 4 q 4 q 4 q 5 7 7 q + 5 q + ----- + ----- + ---- + ---- + -- + --- + ---- + 7 q t + 6 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 13 5 > 6 q t + 7 q t + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 15 5 15 6 17 6 19 7 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a250

K11a249 K11a251