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

K11a254 K11a256

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X12,6,13,5 X20,8,21,7 X16,9,17,10 X18,11,19,12 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, -5, 9, -6, 10, -4, 11, -7}

DT (Dowker-Thistlethwaite) Code:

6 8 12 20 16 18 22 4 10 2 14

Alexander Polynomial:

- t-4 + 6t-3 - 17t-2 + 30t-1 - 35 + 30t - 17t2 + 6t3 - t4

Conway Polynomial:

1 - z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a79, ...}

Determinant and Signature:

{143, -2}

Jones Polynomial:

- q-8 + 4q-7 - 9q-6 + 15q-5 - 20q-4 + 23q-3 - 23q-2 + 20q-1 - 14 + 9q - 4q2 + q3

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

{K11a79, ...}

A2 (sl(3)) Invariant:

- q-24 + q-22 - 2q-18 + 4q-16 - 3q-14 + 2q-12 + q-10 - 3q-8 + 4q-6 - 5q-4 + 4q-2 - q2 + 3q4 - 2q6 + q8

HOMFLY-PT Polynomial:

2 + 3z2 + 3z4 + z6 - 3a2 - 9a2z2 - 10a2z4 - 5a2z6 - a2z8 + 3a4 + 8a4z2 + 7a4z4 + 2a4z6 - a6 - 2a6z2 - a6z4

Kauffman Polynomial:

- 2a-2z4 + a-2z6 + 3a-1z3 - 9a-1z5 + 4a-1z7 + 2 - 8z2 + 19z4 - 22z6 + 8z8 + az - 6az3 + 15az5 - 19az7 + 8az9 + 3a2 - 22a2z2 + 50a2z4 - 42a2z6 + 9a2z8 + 3a2z10 + a3z - 13a3z3 + 38a3z5 - 40a3z7 + 16a3z9 + 3a4 - 20a4z2 + 44a4z4 - 39a4z6 + 11a4z8 + 3a4z10 - a5z + 2a5z3 + a5z5 - 9a5z7 + 8a5z9 + a6 - 5a6z2 + 10a6z4 - 16a6z6 + 10a6z8 - a7z + 5a7z3 - 12a7z5 + 8a7z7 + a8z2 - 5a8z4 + 4a8z6 - a9z3 + a9z5

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

{0, -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=-2 is the signature of 11255. 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 = 7                       1 j = 5                     3   j = 3                   6 1   j = 1                 8 3     j = -1               12 6       j = -3             12 9         j = -5           11 11           j = -7         9 12             j = -9       6 11               j = -11     3 9                 j = -13   1 6                   j = -15   3                     j = -17 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, 255]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 255]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[16, 9, 17, 10], X[18, 11, 19, 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, 255]]
Out[4]=	GaussCode[1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, 7, -11, 8, -5, 9, -6, 10, > -4, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 255]]
Out[5]=	DTCode[6, 8, 12, 20, 16, 18, 22, 4, 10, 2, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 255]][t]
Out[6]=	-4 6 17 30 2 3 4 -35 - t + -- - -- + -- + 30 t - 17 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 255]][z]
Out[7]=	4 6 8 1 - z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 79], Knot[11, Alternating, 255]}
In[9]:=	{KnotDet[Knot[11, Alternating, 255]], KnotSignature[Knot[11, Alternating, 255]]}
Out[9]=	{143, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 255]][q]
Out[10]=	-8 4 9 15 20 23 23 20 2 3 -14 - q + -- - -- + -- - -- + -- - -- + -- + 9 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 79], Knot[11, Alternating, 255]}
In[12]:=	A2Invariant[Knot[11, Alternating, 255]][q]
Out[12]=	-24 -22 2 4 3 2 -10 3 4 5 4 2 4 -q + q - --- + --- - --- + --- + q - -- + -- - -- + -- - q + 3 q - 18 16 14 12 8 6 4 2 q q q q q q q q 6 8 > 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 255]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 4 2 - 3 a + 3 a - a + 3 z - 9 a z + 8 a z - 2 a z + 3 z - 10 a z + 4 4 6 4 6 2 6 4 6 2 8 > 7 a z - a z + z - 5 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 255]][a, z]
Out[14]=	2 4 6 3 5 7 2 2 2 4 2 2 + 3 a + 3 a + a + a z + a z - a z - a z - 8 z - 22 a z - 20 a z - 3 6 2 8 2 3 z 3 3 3 5 3 7 3 9 3 > 5 a z + a z + ---- - 6 a z - 13 a z + 2 a z + 5 a z - a z + a 4 5 4 2 z 2 4 4 4 6 4 8 4 9 z 5 > 19 z - ---- + 50 a z + 44 a z + 10 a z - 5 a z - ---- + 15 a z + 2 a a 6 3 5 5 5 7 5 9 5 6 z 2 6 4 6 > 38 a z + a z - 12 a z + a z - 22 z + -- - 42 a z - 39 a z - 2 a 7 6 6 8 6 4 z 7 3 7 5 7 7 7 8 > 16 a z + 4 a z + ---- - 19 a z - 40 a z - 9 a z + 8 a z + 8 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 9 a z + 11 a z + 10 a z + 8 a z + 16 a z + 8 a z + 3 a z + 4 10 > 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 255]], Vassiliev[3][Knot[11, Alternating, 255]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 255]][q, t]
Out[16]=	9 12 1 3 1 6 3 9 6 11 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 9 12 11 11 12 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 8 q t + 3 q t + 6 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a255

K11a254 K11a256