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

K11a296 K11a298

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 22 20 18 2 8 12 4 14

Alexander Polynomial:

2t-3 - 15t-2 + 42t-1 - 57 + 42t - 15t2 + 2t3

Conway Polynomial:

1 - 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{175, -2}

Jones Polynomial:

- q-8 + 5q-7 - 12q-6 + 19q-5 - 25q-4 + 29q-3 - 28q-2 + 24q-1 - 17 + 10q - 4q2 + q3

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

{K11a125, ...}

A2 (sl(3)) Invariant:

- q-24 + 3q-22 - 3q-20 - 3q-18 + 6q-16 - 4q-14 + 4q-12 + q-10 - 3q-8 + 3q-6 - 6q-4 + 5q-2 - 2q2 + 5q4 - 2q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 + 2 - 2z4 - 3a2 - 4a2z2 - a2z4 + a2z6 + 3a4 + 3a4z2 + a4z4 + a4z6 - a6 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 4a-1z3 - 8a-1z5 + 4a-1z7 + 2 - 6z2 + 13z4 - 17z6 + 8z8 + 2az - 5az3 + 10az5 - 16az7 + 9az9 + 3a2 - 17a2z2 + 38a2z4 - 40a2z6 + 11a2z8 + 4a2z10 + 4a3z - 12a3z3 + 27a3z5 - 41a3z7 + 21a3z9 + 3a4 - 13a4z2 + 37a4z4 - 51a4z6 + 19a4z8 + 4a4z10 + 2a5z + a5z3 - 7a5z5 - 9a5z7 + 12a5z9 + a6 - 3a6z2 + 11a6z4 - 24a6z6 + 16a6z8 + 4a7z3 - 15a7z5 + 12a7z7 - 3a8z4 + 5a8z6 + 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 11297. 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                   7 1   j = 1                 10 3     j = -1               14 7       j = -3             15 11         j = -5           14 13           j = -7         11 15             j = -9       8 14               j = -11     4 11                 j = -13   1 8                   j = -15   4                     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, 297]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 297]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[22, 8, 1, 7], > X[20, 9, 21, 10], X[18, 12, 19, 11], X[2, 13, 3, 14], X[8, 15, 9, 16], > X[12, 18, 13, 17], X[4, 19, 5, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 297]]
Out[4]=	GaussCode[1, -7, 2, -10, 3, -1, 4, -8, 5, -2, 6, -9, 7, -11, 8, -3, 9, -6, 10, > -5, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 297]]
Out[5]=	DTCode[6, 10, 16, 22, 20, 18, 2, 8, 12, 4, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 297]][t]
Out[6]=	2 15 42 2 3 -57 + -- - -- + -- + 42 t - 15 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 297]][z]
Out[7]=	4 6 1 - 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 297]}
In[9]:=	{KnotDet[Knot[11, Alternating, 297]], KnotSignature[Knot[11, Alternating, 297]]}
Out[9]=	{175, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 297]][q]
Out[10]=	-8 5 12 19 25 29 28 24 2 3 -17 - q + -- - -- + -- - -- + -- - -- + -- + 10 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, 125], Knot[11, Alternating, 297]}
In[12]:=	A2Invariant[Knot[11, Alternating, 297]][q]
Out[12]=	-24 3 3 3 6 4 4 -10 3 3 6 5 2 -q + --- - --- - --- + --- - --- + --- + q - -- + -- - -- + -- - 2 q + 22 20 18 16 14 12 8 6 4 2 q q q q q q q q q q 4 6 8 10 > 5 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 297]][a, z]
Out[13]=	2 2 4 6 z 2 2 4 2 4 2 4 4 4 6 4 2 - 3 a + 3 a - a + -- - 4 a z + 3 a z - 2 z - a z + a z - a z + 2 a 2 6 4 6 > a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 297]][a, z]
Out[14]=	2 2 4 6 3 5 2 z 2 2 2 + 3 a + 3 a + a + 2 a z + 4 a z + 2 a z - 6 z + -- - 17 a z - 2 a 3 4 2 6 2 4 z 3 3 3 5 3 7 3 4 > 13 a z - 3 a z + ---- - 5 a z - 12 a z + a z + 4 a z + 13 z - a 4 5 2 z 2 4 4 4 6 4 8 4 8 z 5 > ---- + 38 a z + 37 a z + 11 a z - 3 a z - ---- + 10 a z + 2 a a 6 3 5 5 5 7 5 9 5 6 z 2 6 4 6 > 27 a z - 7 a z - 15 a z + a z - 17 z + -- - 40 a z - 51 a z - 2 a 7 6 6 8 6 4 z 7 3 7 5 7 7 7 > 24 a z + 5 a z + ---- - 16 a z - 41 a z - 9 a z + 12 a z + a 8 2 8 4 8 6 8 9 3 9 5 9 > 8 z + 11 a z + 19 a z + 16 a z + 9 a z + 21 a z + 12 a z + 2 10 4 10 > 4 a z + 4 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 297]], Vassiliev[3][Knot[11, Alternating, 297]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 297]][q, t]
Out[16]=	11 14 1 4 1 8 4 11 8 14 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 11 15 14 13 15 7 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 10 q t + 3 q t + 7 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 K11a297

K11a296 K11a298