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

K11a259 K11a261

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 14 2 20 18 4 22 12 10 16

Alexander Polynomial:

- 2t-3 + 9t-2 - 15t-1 + 17 - 15t + 9t2 - 2t3

Conway Polynomial:

1 + 3z2 - 3z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{69, 4}

Jones Polynomial:

1 - 2q + 5q2 - 7q3 + 9q4 - 11q5 + 11q6 - 9q7 + 7q8 - 4q9 + 2q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 + q4 + 2q6 - q8 + 2q10 - 2q12 - q14 - 2q18 + 2q20 + 2q24 + 2q26 - q28 - q32 - q34

HOMFLY-PT Polynomial:

- 2a-10 - a-10z2 + 4a-8 + 6a-8z2 + 2a-8z4 - 2a-6 - 3a-6z2 - 3a-6z4 - a-6z6 - a-4 - 2a-4z2 - 3a-4z4 - a-4z6 + 2a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

a-13z - 3a-13z3 + a-13z5 + a-12z2 - 5a-12z4 + 2a-12z6 - a-11z + 5a-11z3 - 8a-11z5 + 3a-11z7 + 2a-10 - 13a-10z2 + 23a-10z4 - 15a-10z6 + 4a-10z8 + a-9z - 2a-9z3 + 13a-9z5 - 10a-9z7 + 3a-9z9 + 4a-8 - 25a-8z2 + 41a-8z4 - 19a-8z6 + 2a-8z8 + a-8z10 + 7a-7z - 25a-7z3 + 35a-7z5 - 20a-7z7 + 5a-7z9 + 2a-6 - 11a-6z2 + 12a-6z4 - 6a-6z6 + a-6z10 + 5a-5z - 12a-5z3 + 7a-5z5 - 5a-5z7 + 2a-5z9 - a-4 + 5a-4z2 - 5a-4z4 - 3a-4z6 + 2a-4z8 + a-3z + 3a-3z3 - 6a-3z5 + 2a-3z7 - 2a-2 + 5a-2z2 - 4a-2z4 + a-2z6

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

{3, 9}

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 11260. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     1   j = 19                   3 1   j = 17                 4 1     j = 15               5 3       j = 13             6 4         j = 11           5 5           j = 9         4 6             j = 7       3 5               j = 5     2 4                 j = 3   1 4                   j = 1   1                     j = -1 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, 260]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 260]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[20, 10, 21, 9], X[18, 12, 19, 11], X[4, 14, 5, 13], X[22, 15, 1, 16], > X[12, 18, 13, 17], X[10, 20, 11, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 260]]
Out[4]=	GaussCode[1, -4, 2, -7, 3, -1, 4, -2, 5, -10, 6, -9, 7, -3, 8, -11, 9, -6, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 260]]
Out[5]=	DTCode[6, 8, 14, 2, 20, 18, 4, 22, 12, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 260]][t]
Out[6]=	2 9 15 2 3 17 - -- + -- - -- - 15 t + 9 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 260]][z]
Out[7]=	2 4 6 1 + 3 z - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 260]}
In[9]:=	{KnotDet[Knot[11, Alternating, 260]], KnotSignature[Knot[11, Alternating, 260]]}
Out[9]=	{69, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 260]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 1 - 2 q + 5 q - 7 q + 9 q - 11 q + 11 q - 9 q + 7 q - 4 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 260]}
In[12]:=	A2Invariant[Knot[11, Alternating, 260]][q]
Out[12]=	4 6 8 10 12 14 18 20 24 26 1 + q + 2 q - q + 2 q - 2 q - q - 2 q + 2 q + 2 q + 2 q - 28 32 34 > q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 260]][a, z]
Out[13]=	2 2 2 2 2 4 4 -2 4 2 -4 2 z 6 z 3 z 2 z 3 z 2 z 3 z --- + -- - -- - a + -- - --- + ---- - ---- - ---- + ---- + ---- - ---- - 10 8 6 2 10 8 6 4 2 8 6 a a a a a a a a a a a 4 4 6 6 3 z z z z > ---- + -- - -- - -- 4 2 6 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 260]][a, z]
Out[14]=	2 2 2 4 2 -4 2 z z z 7 z 5 z z z 13 z --- + -- + -- - a - -- + --- - --- + -- + --- + --- + -- + --- - ----- - 10 8 6 2 13 11 9 7 5 3 12 10 a a a a a a a a a a a a 2 2 2 2 3 3 3 3 3 3 25 z 11 z 5 z 5 z 3 z 5 z 2 z 25 z 12 z 3 z > ----- - ----- + ---- + ---- - ---- + ---- - ---- - ----- - ----- + ---- - 8 6 4 2 13 11 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 4 5 5 5 5 5 z 23 z 41 z 12 z 5 z 4 z z 8 z 13 z 35 z > ---- + ----- + ----- + ----- - ---- - ---- + --- - ---- + ----- + ----- + 12 10 8 6 4 2 13 11 9 7 a a a a a a a a a a 5 5 6 6 6 6 6 6 7 7 7 z 6 z 2 z 15 z 19 z 6 z 3 z z 3 z 10 z > ---- - ---- + ---- - ----- - ----- - ---- - ---- + -- + ---- - ----- - 5 3 12 10 8 6 4 2 11 9 a a a a a a a a a a 7 7 7 8 8 8 9 9 9 10 10 20 z 5 z 2 z 4 z 2 z 2 z 3 z 5 z 2 z z z > ----- - ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 7 5 3 10 8 4 9 7 5 8 6 a a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 260]], Vassiliev[3][Knot[11, Alternating, 260]]}
Out[15]=	{3, 9}
In[16]:=	Kh[Knot[11, Alternating, 260]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 4 q t + 3 q t + 5 q t + 4 q t + 6 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 5 q t + 5 q t + 6 q t + 4 q t + 5 q t + 3 q t + 17 6 17 7 19 7 19 8 21 8 23 9 > 4 q t + q t + 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a260

K11a259 K11a261