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

K11a256 K11a258

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 14 2 16 18 20 4 22 12 10

Alexander Polynomial:

t-4 - 5t-3 + 12t-2 - 19t-1 + 23 - 19t + 12t2 - 5t3 + t4

Conway Polynomial:

1 + 2z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{10118, ...}

Determinant and Signature:

{97, 0}

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 10q-3 + 13q-2 - 15q-1 + 16 - 13q + 10q2 - 6q3 + 3q4 - q5

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

{K11a110, ...}

A2 (sl(3)) Invariant:

q-18 + q-12 - 3q-10 + q-8 - 2q-6 - q-4 + 3q-2 - 1 + 5q2 - q4 + q6 + q8 - 2q10 + q12 - q14

HOMFLY-PT Polynomial:

- 2a-2 - 5a-2z2 - 4a-2z4 - a-2z6 + 7 + 15z2 + 14z4 + 6z6 + z8 - 6a2 - 13a2z2 - 9a2z4 - 2a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + a-4z2 - 6a-4z4 + 3a-4z6 - 2a-3z + 7a-3z3 - 11a-3z5 + 5a-3z7 + 2a-2 - 11a-2z2 + 20a-2z4 - 16a-2z6 + 6a-2z8 - 4a-1z + 10a-1z3 - 6a-1z7 + 4a-1z9 + 7 - 31z2 + 55z4 - 36z6 + 9z8 + z10 - 4az + 2az3 + 14az5 - 17az7 + 7az9 + 6a2 - 25a2z2 + 38a2z4 - 28a2z6 + 7a2z8 + a2z10 - 4a3z + 8a3z3 - 7a3z5 - 3a3z7 + 3a3z9 + 2a4 - 4a4z2 + 6a4z4 - 10a4z6 + 4a4z8 - 2a5z + 7a5z3 - 9a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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

{0, 2}

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 11257. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     2   j = 7                   4 1   j = 5                 6 2     j = 3               7 4       j = 1             9 6         j = -1           7 8           j = -3         6 8             j = -5       4 7               j = -7     2 6                 j = -9   1 4                   j = -11   2                     j = -13 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, 257]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 257]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[16, 10, 17, 9], X[18, 11, 19, 12], X[20, 13, 21, 14], X[4, 15, 5, 16], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 257]]
Out[4]=	GaussCode[1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 257]]
Out[5]=	DTCode[6, 8, 14, 2, 16, 18, 20, 4, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 257]][t]
Out[6]=	-4 5 12 19 2 3 4 23 + t - -- + -- - -- - 19 t + 12 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 257]][z]
Out[7]=	4 6 8 1 + 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 118], Knot[11, Alternating, 257]}
In[9]:=	{KnotDet[Knot[11, Alternating, 257]], KnotSignature[Knot[11, Alternating, 257]]}
Out[9]=	{97, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 257]][q]
Out[10]=	-6 3 6 10 13 15 2 3 4 5 16 + q - -- + -- - -- + -- - -- - 13 q + 10 q - 6 q + 3 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 110], Knot[11, Alternating, 257]}
In[12]:=	A2Invariant[Knot[11, Alternating, 257]][q]
Out[12]=	-18 -12 3 -8 2 -4 3 2 4 6 8 10 -1 + q + q - --- + q - -- - q + -- + 5 q - q + q + q - 2 q + 10 6 2 q q q 12 14 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 257]][a, z]
Out[13]=	2 4 2 2 4 2 5 z 2 2 4 2 4 4 z 7 - -- - 6 a + 2 a + 15 z - ---- - 13 a z + 3 a z + 14 z - ---- - 2 2 2 a a a 6 2 4 4 4 6 z 2 6 8 > 9 a z + a z + 6 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 257]][a, z]
Out[14]=	2 2 2 4 2 z 4 z 3 5 2 z 7 + -- + 6 a + 2 a - --- - --- - 4 a z - 4 a z - 2 a z - 31 z + -- - 2 3 a 4 a a a 2 3 3 3 11 z 2 2 4 2 6 2 2 z 7 z 10 z 3 > ----- - 25 a z - 4 a z + 2 a z - ---- + ---- + ----- + 2 a z + 2 5 3 a a a a 4 4 3 3 5 3 4 6 z 20 z 2 4 4 4 6 4 > 8 a z + 7 a z + 55 z - ---- + ----- + 38 a z + 6 a z - 3 a z + 4 2 a a 5 5 6 6 z 11 z 5 3 5 5 5 6 3 z 16 z > -- - ----- + 14 a z - 7 a z - 9 a z - 36 z + ---- - ----- - 5 3 4 2 a a a a 7 7 2 6 4 6 6 6 5 z 6 z 7 3 7 5 7 > 28 a z - 10 a z + a z + ---- - ---- - 17 a z - 3 a z + 3 a z + 3 a a 8 9 8 6 z 2 8 4 8 4 z 9 3 9 10 2 10 > 9 z + ---- + 7 a z + 4 a z + ---- + 7 a z + 3 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 257]], Vassiliev[3][Knot[11, Alternating, 257]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 257]][q, t]
Out[16]=	8 1 2 1 4 2 6 4 7 - + 9 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 6 8 7 3 3 2 5 2 5 3 > ----- + ---- + --- + 6 q t + 7 q t + 4 q t + 6 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a257

K11a256 K11a258