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

K11a255 K11a257

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X12,6,13,5 X20,8,21,7 X18,9,19,10 X16,11,17,12 X22,13,1,14 X4,16,5,15 X10,17,11,18 X2,19,3,20 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 12 20 18 16 22 4 10 2 14

Alexander Polynomial:

- 2t-3 + 12t-2 - 31t-1 + 43 - 31t + 12t2 - 2t3

Conway Polynomial:

1 - z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{133, 0}

Jones Polynomial:

- q-7 + 4q-6 - 8q-5 + 13q-4 - 18q-3 + 21q-2 - 21q-1 + 19 - 14q + 9q2 - 4q3 + q4

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

{...}

A2 (sl(3)) Invariant:

- q-22 + q-20 + 2q-18 - 3q-16 + 2q-14 - 3q-10 + 4q-8 - 2q-6 + 2q-4 - 2 + 4q2 - 4q4 + 2q6 + 2q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2 + a-2z2 + a-2z4 - 1 - 3z2 - 2z4 - z6 + a2 - a2z4 - a2z6 + 2a4z2 + 2a4z4 - a6z2

Kauffman Polynomial:

a-4z4 - a-3z3 + 4a-3z5 - a-2 + 3a-2z2 - 8a-2z4 + 9a-2z6 - a-1z + 7a-1z3 - 17a-1z5 + 13a-1z7 - 1 + 4z2 + 2z4 - 18z6 + 13z8 - az + 7az3 - 9az5 - 9az7 + 9az9 - a2 - 5a2z2 + 32a2z4 - 43a2z6 + 11a2z8 + 3a2z10 - 8a3z3 + 34a3z5 - 43a3z7 + 15a3z9 - 10a4z2 + 35a4z4 - 30a4z6 + 2a4z8 + 3a4z10 - 5a5z3 + 19a5z5 - 20a5z7 + 6a5z9 - 4a6z2 + 14a6z4 - 14a6z6 + 4a6z8 + 2a7z3 - 3a7z5 + a7z7

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 11256. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                       1 j = 7                     3   j = 5                   6 1   j = 3                 8 3     j = 1               11 6       j = -1             11 9         j = -3           10 10           j = -5         8 11             j = -7       5 10               j = -9     3 8                 j = -11   1 5                   j = -13   3                     j = -15 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, 256]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 256]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[18, 9, 19, 10], X[16, 11, 17, 12], X[22, 13, 1, 14], X[4, 16, 5, 15], > X[10, 17, 11, 18], X[2, 19, 3, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 256]]
Out[4]=	GaussCode[1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, 7, -11, 8, -6, 9, -5, 10, > -4, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 256]]
Out[5]=	DTCode[6, 8, 12, 20, 18, 16, 22, 4, 10, 2, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 256]][t]
Out[6]=	2 12 31 2 3 43 - -- + -- - -- - 31 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 256]][z]
Out[7]=	2 6 1 - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 256]}
In[9]:=	{KnotDet[Knot[11, Alternating, 256]], KnotSignature[Knot[11, Alternating, 256]]}
Out[9]=	{133, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 256]][q]
Out[10]=	-7 4 8 13 18 21 21 2 3 4 19 - q + -- - -- + -- - -- + -- - -- - 14 q + 9 q - 4 q + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 256]}
In[12]:=	A2Invariant[Knot[11, Alternating, 256]][q]
Out[12]=	-22 -20 2 3 2 3 4 2 2 2 4 6 -2 - q + q + --- - --- + --- - --- + -- - -- + -- + 4 q - 4 q + 2 q + 18 16 14 10 8 6 4 q q q q q q q 8 10 12 > 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 256]][a, z]
Out[13]=	2 4 -2 2 2 z 4 2 6 2 4 z 2 4 4 4 -1 + a + a - 3 z + -- + 2 a z - a z - 2 z + -- - a z + 2 a z - 2 2 a a 6 2 6 > z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 256]][a, z]
Out[14]=	2 3 -2 2 z 2 3 z 2 2 4 2 6 2 z -1 - a - a - - - a z + 4 z + ---- - 5 a z - 10 a z - 4 a z - -- + a 2 3 a a 3 4 4 7 z 3 3 3 5 3 7 3 4 z 8 z 2 4 > ---- + 7 a z - 8 a z - 5 a z + 2 a z + 2 z + -- - ---- + 32 a z + a 4 2 a a 5 5 4 4 6 4 4 z 17 z 5 3 5 5 5 > 35 a z + 14 a z + ---- - ----- - 9 a z + 34 a z + 19 a z - 3 a a 6 7 7 5 6 9 z 2 6 4 6 6 6 13 z 7 > 3 a z - 18 z + ---- - 43 a z - 30 a z - 14 a z + ----- - 9 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 43 a z - 20 a z + a z + 13 z + 11 a z + 2 a z + 4 a z + 9 3 9 5 9 2 10 4 10 > 9 a z + 15 a z + 6 a z + 3 a z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 256]], Vassiliev[3][Knot[11, Alternating, 256]]}
Out[15]=	{-1, 0}
In[16]:=	Kh[Knot[11, Alternating, 256]][q, t]
Out[16]=	9 1 3 1 5 3 8 5 10 - + 11 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 8 11 10 10 11 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 6 q t + 8 q t + 3 q t + 6 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a256

K11a255 K11a257