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

K11a192 K11a194

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 18 20 16 2 22 10 8 6

Alexander Polynomial:

2t-3 - 10t-2 + 22t-1 - 27 + 22t - 10t2 + 2t3

Conway Polynomial:

1 + 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{95, -2}

Jones Polynomial:

q-9 - 3q-8 + 6q-7 - 10q-6 + 13q-5 - 15q-4 + 15q-3 - 13q-2 + 10q-1 - 5 + 3q - q2

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-24 + 2q-22 - 2q-20 - q-18 + q-16 - 3q-14 + 2q-12 - q-10 + q-8 + 2q-6 - 2q-4 + 4q-2 + q4 - q6

HOMFLY-PT Polynomial:

- 2z2 - z4 + 2a2 + 4a2z2 + 3a2z4 + a2z6 + a4z2 + 2a4z4 + a4z6 - 2a6 - 4a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

- 2a-1z3 + a-1z5 + 4z2 - 7z4 + 3z6 + 3az3 - 7az5 + 4az7 - 2a2 + 5a2z2 - 5a2z6 + 4a2z8 + 3a3z - 4a3z3 + 4a3z5 - 3a3z7 + 3a3z9 - 6a4z2 + 15a4z4 - 13a4z6 + 5a4z8 + a4z10 + 5a5z - 17a5z3 + 20a5z5 - 14a5z7 + 6a5z9 + 2a6 - 11a6z2 + 17a6z4 - 16a6z6 + 5a6z8 + a6z10 + a7z - a7z3 - a7z5 - 4a7z7 + 3a7z9 + a8 - 2a8z2 + 6a8z4 - 10a8z6 + 4a8z8 - a9z + 7a9z3 - 9a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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=-2 is the signature of 11193. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     2   j = 1                   3 1   j = -1                 7 2     j = -3               7 4       j = -5             8 6         j = -7           7 7           j = -9         6 8             j = -11       4 7               j = -13     2 6                 j = -15   1 4                   j = -17   2                     j = -19 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, 193]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 193]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[18, 7, 19, 8], > X[20, 9, 21, 10], X[16, 11, 17, 12], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[10, 17, 11, 18], X[8, 19, 9, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 193]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -6, 9, -4, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 193]]
Out[5]=	DTCode[4, 12, 14, 18, 20, 16, 2, 22, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 193]][t]
Out[6]=	2 10 22 2 3 -27 + -- - -- + -- + 22 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 193]][z]
Out[7]=	4 6 1 + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 193]}
In[9]:=	{KnotDet[Knot[11, Alternating, 193]], KnotSignature[Knot[11, Alternating, 193]]}
Out[9]=	{95, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 193]][q]
Out[10]=	-9 3 6 10 13 15 15 13 10 2 -5 + q - -- + -- - -- + -- - -- + -- - -- + -- + 3 q - q 8 7 6 5 4 3 2 q 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, 193]}
In[12]:=	A2Invariant[Knot[11, Alternating, 193]][q]
Out[12]=	-28 -24 2 2 -18 -16 3 2 -10 -8 2 2 4 q - q + --- - --- - q + q - --- + --- - q + q + -- - -- + -- + 22 20 14 12 6 4 2 q q q q q q q 4 6 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 193]][a, z]
Out[13]=	2 6 8 2 2 2 4 2 6 2 8 2 4 2 4 2 a - 2 a + a - 2 z + 4 a z + a z - 4 a z + a z - z + 3 a z + 4 4 6 4 2 6 4 6 > 2 a z - 2 a z + a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 193]][a, z]
Out[14]=	2 6 8 3 5 7 9 2 2 2 4 2 -2 a + 2 a + a + 3 a z + 5 a z + a z - a z + 4 z + 5 a z - 6 a z - 3 6 2 8 2 10 2 2 z 3 3 3 5 3 > 11 a z - 2 a z + 2 a z - ---- + 3 a z - 4 a z - 17 a z - a 5 7 3 9 3 4 4 4 6 4 8 4 10 4 z > a z + 7 a z - 7 z + 15 a z + 17 a z + 6 a z - 3 a z + -- - a 5 3 5 5 5 7 5 9 5 6 2 6 4 6 > 7 a z + 4 a z + 20 a z - a z - 9 a z + 3 z - 5 a z - 13 a z - 6 6 8 6 10 6 7 3 7 5 7 7 7 > 16 a z - 10 a z + a z + 4 a z - 3 a z - 14 a z - 4 a z + 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 3 a z + 4 a z + 5 a z + 5 a z + 4 a z + 3 a z + 6 a z + 7 9 4 10 6 10 > 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 193]], Vassiliev[3][Knot[11, Alternating, 193]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 193]][q, t]
Out[16]=	4 7 1 2 1 4 2 6 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 7 6 8 7 7 8 6 7 2 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a193

K11a192 K11a194