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

K11a297 K11a299

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,4,11,3 X18,6,19,5 X2837 X16,10,17,9 X20,12,21,11 X22,14,1,13 X4,16,5,15 X8,18,9,17 X14,20,15,19 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 18 2 16 20 22 4 8 14 12

Alexander Polynomial:

5t-3 - 16t-2 + 28t-1 - 33 + 28t - 16t2 + 5t3

Conway Polynomial:

1 + 9z2 + 14z4 + 5z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{131, 6}

Jones Polynomial:

q3 - 3q4 + 8q5 - 12q6 + 18q7 - 21q8 + 21q9 - 19q10 + 14q11 - 9q12 + 4q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q10 - 2q12 + 3q14 - q16 + q18 + 5q20 - 2q22 + 5q24 - 2q26 - q28 - 5q32 + 2q34 - 2q36 + 2q40 - q42

HOMFLY-PT Polynomial:

a-12 - a-12z2 - a-12z4 - 6a-10 - 7a-10z2 + a-10z6 + 6a-8 + 15a-8z2 + 12a-8z4 + 3a-8z6 + 2a-6z2 + 3a-6z4 + a-6z6

Kauffman Polynomial:

- a-17z3 + a-17z5 + a-16z2 - 5a-16z4 + 4a-16z6 - 2a-15z + 7a-15z3 - 13a-15z5 + 8a-15z7 - 2a-14z2 + 6a-14z4 - 13a-14z6 + 9a-14z8 - a-13z + 8a-13z3 - 11a-13z5 - a-13z7 + 6a-13z9 + a-12 - 4a-12z2 + 19a-12z4 - 27a-12z6 + 11a-12z8 + 2a-12z10 - 5a-11z + 8a-11z3 + 6a-11z5 - 19a-11z7 + 11a-11z9 + 6a-10 - 21a-10z2 + 34a-10z4 - 29a-10z6 + 8a-10z8 + 2a-10z10 - 6a-9z + 11a-9z3 - 4a-9z5 - 7a-9z7 + 5a-9z9 + 6a-8 - 18a-8z2 + 23a-8z4 - 18a-8z6 + 6a-8z8 + 3a-7z3 - 7a-7z5 + 3a-7z7 + 2a-6z2 - 3a-6z4 + a-6z6

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

{9, 25}

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=6 is the signature of 11298. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 29                       1 j = 27                     3   j = 25                   6 1   j = 23                 8 3     j = 21               11 6       j = 19             10 8         j = 17           11 11           j = 15         7 10             j = 13       5 11               j = 11     3 7                 j = 9     5                   j = 7 1 3                     j = 5 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, 298]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 298]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[18, 6, 19, 5], X[2, 8, 3, 7], > X[16, 10, 17, 9], X[20, 12, 21, 11], X[22, 14, 1, 13], X[4, 16, 5, 15], > X[8, 18, 9, 17], X[14, 20, 15, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 298]]
Out[4]=	GaussCode[1, -4, 2, -8, 3, -1, 4, -9, 5, -2, 6, -11, 7, -10, 8, -5, 9, -3, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 298]]
Out[5]=	DTCode[6, 10, 18, 2, 16, 20, 22, 4, 8, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 298]][t]
Out[6]=	5 16 28 2 3 -33 + -- - -- + -- + 28 t - 16 t + 5 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 298]][z]
Out[7]=	2 4 6 1 + 9 z + 14 z + 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 298]}
In[9]:=	{KnotDet[Knot[11, Alternating, 298]], KnotSignature[Knot[11, Alternating, 298]]}
Out[9]=	{131, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 298]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 3 q + 8 q - 12 q + 18 q - 21 q + 21 q - 19 q + 14 q - 9 q + 13 14 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 298]}
In[12]:=	A2Invariant[Knot[11, Alternating, 298]][q]
Out[12]=	10 12 14 16 18 20 22 24 26 28 32 q - 2 q + 3 q - q + q + 5 q - 2 q + 5 q - 2 q - q - 5 q + 34 36 40 42 > 2 q - 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 298]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 6 -12 6 6 z 7 z 15 z 2 z z 12 z 3 z z 3 z a - --- + -- - --- - ---- + ----- + ---- - --- + ----- + ---- + --- + ---- + 10 8 12 10 8 6 12 8 6 10 8 a a a a a a a a a a a 6 z > -- 6 a
In[14]:=	Kauffman[Knot[11, Alternating, 298]][a, z]
Out[14]=	2 2 2 2 2 -12 6 6 2 z z 5 z 6 z z 2 z 4 z 21 z 18 z a + --- + -- - --- - --- - --- - --- + --- - ---- - ---- - ----- - ----- + 10 8 15 13 11 9 16 14 12 10 8 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 2 z z 7 z 8 z 8 z 11 z 3 z 5 z 6 z 19 z > ---- - --- + ---- + ---- + ---- + ----- + ---- - ---- + ---- + ----- + 6 17 15 13 11 9 7 16 14 12 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 34 z 23 z 3 z z 13 z 11 z 6 z 4 z 7 z 4 z > ----- + ----- - ---- + --- - ----- - ----- + ---- - ---- - ---- + ---- - 10 8 6 17 15 13 11 9 7 16 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 13 z 27 z 29 z 18 z z 8 z z 19 z 7 z 3 z > ----- - ----- - ----- - ----- + -- + ---- - --- - ----- - ---- + ---- + 14 12 10 8 6 15 13 11 9 7 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 9 z 11 z 8 z 6 z 6 z 11 z 5 z 2 z 2 z > ---- + ----- + ---- + ---- + ---- + ----- + ---- + ----- + ----- 14 12 10 8 13 11 9 12 10 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 298]], Vassiliev[3][Knot[11, Alternating, 298]]}
Out[15]=	{9, 25}
In[16]:=	Kh[Knot[11, Alternating, 298]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 3 q t + 5 q t + 3 q t + 7 q t + 5 q t + 11 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 7 q t + 10 q t + 11 q t + 11 q t + 10 q t + 8 q t + 21 7 21 8 23 8 23 9 25 9 25 10 > 11 q t + 6 q t + 8 q t + 3 q t + 6 q t + q t + 27 10 29 11 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a298

K11a297 K11a299