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

K11a197 K11a199

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,4,13,3 X16,5,17,6 X14,7,15,8 X18,10,19,9 X2,12,3,11 X22,13,1,14 X6,15,7,16 X20,18,21,17 X10,20,11,19 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 14 18 2 22 6 20 10 8

Alexander Polynomial:

2t-3 - 11t-2 + 27t-1 - 35 + 27t - 11t2 + 2t3

Conway Polynomial:

1 + z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{10121, K11a41, K11a183, K11a331, ...}

Determinant and Signature:

{115, 2}

Jones Polynomial:

- q-4 + 3q-3 - 6q-2 + 11q-1 - 15 + 18q - 18q2 + 17q3 - 13q4 + 8q5 - 4q6 + q7

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

{...}

A2 (sl(3)) Invariant:

- q-12 + q-10 - q-8 - q-6 + 4q-4 - 2q-2 + 2 + 2q2 - 2q4 + 3q6 - 3q8 + 2q10 - 2q14 + 3q16 - 2q18 - q20 + q22

HOMFLY-PT Polynomial:

a-6z2 - 2a-4z2 - 2a-4z4 - a-2 - a-2z2 + a-2z4 + a-2z6 + 3 + 5z2 + 3z4 + z6 - a2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 2a-7z3 + 4a-7z5 + 2a-6z2 - 7a-6z4 + 8a-6z6 - a-5z + 6a-5z3 - 14a-5z5 + 11a-5z7 + a-4z2 + a-4z4 - 12a-4z6 + 10a-4z8 - 5a-3z + 24a-3z3 - 32a-3z5 + 6a-3z7 + 5a-3z9 + a-2 - 12a-2z2 + 38a-2z4 - 47a-2z6 + 16a-2z8 + a-2z10 - 8a-1z + 25a-1z3 - 15a-1z5 - 12a-1z7 + 8a-1z9 + 3 - 19z2 + 45z4 - 39z6 + 9z8 + z10 - 6az + 14az3 - 5az5 - 6az7 + 3az9 + a2 - 8a2z2 + 16a2z4 - 12a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

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

{1, 0}

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 11198. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 8 3     j = 7               9 5       j = 5             9 8         j = 3           9 9           j = 1         7 10             j = -1       4 8               j = -3     2 7                 j = -5   1 4                   j = -7   2                     j = -9 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, 198]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 198]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[16, 5, 17, 6], X[14, 7, 15, 8], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[6, 15, 7, 16], > X[20, 18, 21, 17], X[10, 20, 11, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 198]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -5, 10, > -9, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 198]]
Out[5]=	DTCode[4, 12, 16, 14, 18, 2, 22, 6, 20, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 198]][t]
Out[6]=	2 11 27 2 3 -35 + -- - -- + -- + 27 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 198]][z]
Out[7]=	2 4 6 1 + z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 121], Knot[11, Alternating, 41], Knot[11, Alternating, 183], > Knot[11, Alternating, 198], Knot[11, Alternating, 331]}
In[9]:=	{KnotDet[Knot[11, Alternating, 198]], KnotSignature[Knot[11, Alternating, 198]]}
Out[9]=	{115, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 198]][q]
Out[10]=	-4 3 6 11 2 3 4 5 6 7 -15 - q + -- - -- + -- + 18 q - 18 q + 17 q - 13 q + 8 q - 4 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 198]}
In[12]:=	A2Invariant[Knot[11, Alternating, 198]][q]
Out[12]=	-12 -10 -8 -6 4 2 2 4 6 8 10 2 - q + q - q - q + -- - -- + 2 q - 2 q + 3 q - 3 q + 2 q - 4 2 q q 14 16 18 20 22 > 2 q + 3 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 198]][a, z]
Out[13]=	2 2 2 4 4 -2 2 2 z 2 z z 2 2 4 2 z z 2 4 3 - a - a + 5 z + -- - ---- - -- - 2 a z + 3 z - ---- + -- - a z + 6 4 2 4 2 a a a a a 6 6 z > z + -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 198]][a, z]
Out[14]=	2 2 2 -2 2 z 5 z 8 z 3 2 2 z z 12 z 3 + a + a - -- - --- - --- - 6 a z - 2 a z - 19 z + ---- + -- - ----- - 5 3 a 6 4 2 a a a a a 3 3 3 3 4 2 2 2 z 6 z 24 z 25 z 3 3 3 4 z > 8 a z - ---- + ---- + ----- + ----- + 14 a z + 5 a z + 45 z + -- - 7 5 3 a 8 a a a a 4 4 4 5 5 5 5 7 z z 38 z 2 4 4 z 14 z 32 z 15 z 5 > ---- + -- + ----- + 16 a z + ---- - ----- - ----- - ----- - 5 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 3 5 6 8 z 12 z 47 z 2 6 11 z 6 z 12 z > 4 a z - 39 z + ---- - ----- - ----- - 12 a z + ----- + ---- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 10 z 16 z 2 8 5 z 8 z 9 > 6 a z + a z + 9 z + ----- + ----- + 3 a z + ---- + ---- + 3 a z + 4 2 3 a a a a 10 10 z > z + --- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 198]], Vassiliev[3][Knot[11, Alternating, 198]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 198]][q, t]
Out[16]=	3 1 2 1 4 2 7 4 8 10 q + 9 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 7 q 3 5 5 2 7 2 7 3 9 3 9 4 > --- + 9 q t + 9 q t + 8 q t + 9 q t + 5 q t + 8 q t + 3 q t + t 11 4 11 5 13 5 15 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a198

K11a197 K11a199