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

K11a289 K11a291

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X16,5,17,6 X20,8,21,7 X18,9,19,10 X2,11,3,12 X8,13,9,14 X22,16,1,15 X4,17,5,18 X12,19,13,20 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 20 18 2 8 22 4 12 14

Alexander Polynomial:

- 3t-3 + 15t-2 - 32t-1 + 41 - 32t + 15t2 - 3t3

Conway Polynomial:

1 + z2 - 3z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{K11a100, ...}

Determinant and Signature:

{141, 0}

Jones Polynomial:

- q-7 + 4q-6 - 9q-5 + 14q-4 - 19q-3 + 23q-2 - 22q-1 + 20 - 15q + 9q2 - 4q3 + q4

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

{...}

A2 (sl(3)) Invariant:

- q-22 + 2q-18 - 3q-16 + 2q-14 - 3q-10 + 5q-8 - 2q-6 + 3q-4 - 2 + 4q2 - 4q4 + 2q6 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 + 1 + z2 - z4 - z6 - a2 - 6a2z2 - 6a2z4 - 2a2z6 + 2a4 + 6a4z2 + 3a4z4 - a6 - a6z2

Kauffman Polynomial:

a-4z4 - a-3z3 + 4a-3z5 + 2a-2z2 - 7a-2z4 + 9a-2z6 - a-1z + 9a-1z3 - 19a-1z5 + 14a-1z7 + 1 - 5z2 + 14z4 - 25z6 + 15z8 - 2az + 12az3 - 14az5 - 9az7 + 10az9 + a2 - 19a2z2 + 54a2z4 - 60a2z6 + 16a2z8 + 3a2z10 - a3z + 2a3z3 + 22a3z5 - 41a3z7 + 16a3z9 + 2a4 - 17a4z2 + 45a4z4 - 39a4z6 + 5a4z8 + 3a4z10 - a5z + 3a5z3 + 10a5z5 - 17a5z7 + 6a5z9 + a6 - 5a6z2 + 13a6z4 - 13a6z6 + 4a6z8 - a7z + 3a7z3 - 3a7z5 + a7z7

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

{1, -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 11290. 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                 9 3     j = 1               11 6       j = -1             12 10         j = -3           11 10           j = -5         8 12             j = -7       6 11               j = -9     3 8                 j = -11   1 6                   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, 290]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 290]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[20, 8, 21, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[8, 13, 9, 14], X[22, 16, 1, 15], > X[4, 17, 5, 18], X[12, 19, 13, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 290]]
Out[4]=	GaussCode[1, -6, 2, -9, 3, -1, 4, -7, 5, -2, 6, -10, 7, -11, 8, -3, 9, -5, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 290]]
Out[5]=	DTCode[6, 10, 16, 20, 18, 2, 8, 22, 4, 12, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 290]][t]
Out[6]=	3 15 32 2 3 41 - -- + -- - -- - 32 t + 15 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 290]][z]
Out[7]=	2 4 6 1 + z - 3 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 100], Knot[11, Alternating, 290]}
In[9]:=	{KnotDet[Knot[11, Alternating, 290]], KnotSignature[Knot[11, Alternating, 290]]}
Out[9]=	{141, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 290]][q]
Out[10]=	-7 4 9 14 19 23 22 2 3 4 20 - q + -- - -- + -- - -- + -- - -- - 15 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, 290]}
In[12]:=	A2Invariant[Knot[11, Alternating, 290]][q]
Out[12]=	-22 2 3 2 3 5 2 3 2 4 6 8 -2 - q + --- - --- + --- - --- + -- - -- + -- + 4 q - 4 q + 2 q + q - 18 16 14 10 8 6 4 q q q q q q q 10 12 > 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 290]][a, z]
Out[13]=	2 4 2 4 6 2 z 2 2 4 2 6 2 4 z 2 4 1 - a + 2 a - a + z + -- - 6 a z + 6 a z - a z - z + -- - 6 a z + 2 2 a a 4 4 6 2 6 > 3 a z - z - 2 a z
In[14]:=	Kauffman[Knot[11, Alternating, 290]][a, z]
Out[14]=	2 2 4 6 z 3 5 7 2 2 z 2 2 1 + a + 2 a + a - - - 2 a z - a z - a z - a z - 5 z + ---- - 19 a z - a 2 a 3 3 4 2 6 2 z 9 z 3 3 3 5 3 7 3 > 17 a z - 5 a z - -- + ---- + 12 a z + 2 a z + 3 a z + 3 a z + 3 a a 4 4 5 5 4 z 7 z 2 4 4 4 6 4 4 z 19 z > 14 z + -- - ---- + 54 a z + 45 a z + 13 a z + ---- - ----- - 4 2 3 a a a a 6 5 3 5 5 5 7 5 6 9 z 2 6 > 14 a z + 22 a z + 10 a z - 3 a z - 25 z + ---- - 60 a z - 2 a 7 4 6 6 6 14 z 7 3 7 5 7 7 7 > 39 a z - 13 a z + ----- - 9 a z - 41 a z - 17 a z + a z + a 8 2 8 4 8 6 8 9 3 9 5 9 > 15 z + 16 a z + 5 a z + 4 a z + 10 a z + 16 a z + 6 a z + 2 10 4 10 > 3 a z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 290]], Vassiliev[3][Knot[11, Alternating, 290]]}
Out[15]=	{1, -2}
In[16]:=	Kh[Knot[11, Alternating, 290]][q, t]
Out[16]=	10 1 3 1 6 3 8 6 11 -- + 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 12 11 10 12 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 6 q t + 9 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 K11a290

K11a289 K11a291