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10119 10121

KnotPlot

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KnotPlot

PD Presentation:

X1627 X5,18,6,19 X13,20,14,1 X11,16,12,17 X3,10,4,11 X7,12,8,13 X9,4,10,5 X15,8,16,9 X19,14,20,15 X17,2,18,3

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 18 12 4 16 20 8 2 14

Minimum Braid Representative: Length is 14, width is 5 Braid index is 5

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 2--3 2 3 / NotAvailable 1

Alexander Polynomial:

8t-2 - 26t-1 + 37 - 26t + 8t2

Conway Polynomial:

1 + 6z2 + 8z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{105, -4}

Jones Polynomial:

q-12 - 4q-11 + 8q-10 - 13q-9 + 16q-8 - 18q-7 + 17q-6 - 13q-5 + 10q-4 - 4q-3 + q-2

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

{...}

A2 (sl(3)) Invariant:

q-38 + q-36 - 3q-34 + q-32 - 5q-28 + 2q-26 - 2q-24 + q-22 + 2q-20 - q-18 + 5q-16 - 2q-14 + 2q-12 + 3q-10 - 3q-8 + q-6

HOMFLY-PT Polynomial:

a4z4 + 3a6 + 7a6z2 + 4a6z4 + 3a8z2 + 3a8z4 - 3a10 - 4a10z2 + a12

Kauffman Polynomial:

a4z4 + 4a5z5 - 3a6 + 7a6z2 - 11a6z4 + 10a6z6 + 2a7z + 5a7z3 - 17a7z5 + 13a7z7 - 3a8z4 - 9a8z6 + 10a8z8 - 4a9z + 26a9z3 - 44a9z5 + 16a9z7 + 3a9z9 + 3a10 - 7a10z2 + 17a10z4 - 33a10z6 + 16a10z8 - 8a11z + 29a11z3 - 33a11z5 + 7a11z7 + 3a11z9 + a12 + a12z2 + 6a12z4 - 13a12z6 + 6a12z8 - 2a13z + 8a13z3 - 10a13z5 + 4a13z7 + a14z2 - 2a14z4 + a14z6

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

{6, -13}

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=-4 is the signature of 10120. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                     1 j = -5                   4 1 j = -7                 6     j = -9               7 4     j = -11             10 6       j = -13           8 7         j = -15         8 10           j = -17       5 8             j = -19     3 8               j = -21   1 5                 j = -23   3                   j = -25 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-34 - 4q-33 + 2q-32 + 15q-31 - 27q-30 - 8q-29 + 70q-28 - 59q-27 - 63q-26 + 158q-25 - 62q-24 - 154q-23 + 226q-22 - 27q-21 - 232q-20 + 239q-19 + 24q-18 - 255q-17 + 192q-16 + 62q-15 - 205q-14 + 109q-13 + 63q-12 - 110q-11 + 38q-10 + 31q-9 - 33q-8 + 7q-7 + 6q-6 - 4q-5 + q-4
3	q-66 - 4q-65 + 2q-64 + 9q-63 + q-62 - 30q-61 - 14q-60 + 71q-59 + 54q-58 - 116q-57 - 158q-56 + 151q-55 + 326q-54 - 125q-53 - 550q-52 + 5q-51 + 782q-50 + 237q-49 - 989q-48 - 555q-47 + 1090q-46 + 950q-45 - 1111q-44 - 1329q-43 + 1013q-42 + 1689q-41 - 855q-40 - 1967q-39 + 626q-38 + 2187q-37 - 384q-36 - 2294q-35 + 112q-34 + 2306q-33 + 163q-32 - 2212q-31 - 400q-30 + 1972q-29 + 615q-28 - 1669q-27 - 704q-26 + 1251q-25 + 741q-24 - 884q-23 - 629q-22 + 519q-21 + 504q-20 - 294q-19 - 314q-18 + 122q-17 + 192q-16 - 57q-15 - 92q-14 + 25q-13 + 37q-12 - 9q-11 - 13q-10 + 3q-9 + 6q-8 - 4q-7 + q-6

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]:=	PD[Knot[10, 120]]
Out[2]=	PD[X[1, 6, 2, 7], X[5, 18, 6, 19], X[13, 20, 14, 1], X[11, 16, 12, 17], > X[3, 10, 4, 11], X[7, 12, 8, 13], X[9, 4, 10, 5], X[15, 8, 16, 9], > X[19, 14, 20, 15], X[17, 2, 18, 3]]
In[3]:=	GaussCode[Knot[10, 120]]
Out[3]=	GaussCode[-1, 10, -5, 7, -2, 1, -6, 8, -7, 5, -4, 6, -3, 9, -8, 4, -10, 2, -9, > 3]
In[4]:=	DTCode[Knot[10, 120]]
Out[4]=	DTCode[6, 10, 18, 12, 4, 16, 20, 8, 2, 14]
In[5]:=	br = BR[Knot[10, 120]]
Out[5]=	BR[5, {-1, -1, -2, 1, 3, 2, -1, -4, -3, -2, -2, -3, -3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 14}
In[7]:=	BraidIndex[Knot[10, 120]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 120]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 120]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 120]][t]
Out[10]=	8 26 2 37 + -- - -- - 26 t + 8 t 2 t t
In[11]:=	Conway[Knot[10, 120]][z]
Out[11]=	2 4 1 + 6 z + 8 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 120]}
In[13]:=	{KnotDet[Knot[10, 120]], KnotSignature[Knot[10, 120]]}
Out[13]=	{105, -4}
In[14]:=	Jones[Knot[10, 120]][q]
Out[14]=	-12 4 8 13 16 18 17 13 10 4 -2 q - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 3 q q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 120]}
In[16]:=	A2Invariant[Knot[10, 120]][q]
Out[16]=	-38 -36 3 -32 5 2 2 -22 2 -18 5 2 q + q - --- + q - --- + --- - --- + q + --- - q + --- - --- + 34 28 26 24 20 16 14 q q q q q q q 2 3 3 -6 > --- + --- - -- + q 12 10 8 q q q
In[17]:=	HOMFLYPT[Knot[10, 120]][a, z]
Out[17]=	6 10 12 6 2 8 2 10 2 4 4 6 4 8 4 3 a - 3 a + a + 7 a z + 3 a z - 4 a z + a z + 4 a z + 3 a z
In[18]:=	Kauffman[Knot[10, 120]][a, z]
Out[18]=	6 10 12 7 9 11 13 6 2 -3 a + 3 a + a + 2 a z - 4 a z - 8 a z - 2 a z + 7 a z - 10 2 12 2 14 2 7 3 9 3 11 3 13 3 > 7 a z + a z + a z + 5 a z + 26 a z + 29 a z + 8 a z + 4 4 6 4 8 4 10 4 12 4 14 4 5 5 > a z - 11 a z - 3 a z + 17 a z + 6 a z - 2 a z + 4 a z - 7 5 9 5 11 5 13 5 6 6 8 6 > 17 a z - 44 a z - 33 a z - 10 a z + 10 a z - 9 a z - 10 6 12 6 14 6 7 7 9 7 11 7 > 33 a z - 13 a z + a z + 13 a z + 16 a z + 7 a z + 13 7 8 8 10 8 12 8 9 9 11 9 > 4 a z + 10 a z + 16 a z + 6 a z + 3 a z + 3 a z
In[19]:=	{Vassiliev[2][Knot[10, 120]], Vassiliev[3][Knot[10, 120]]}
Out[19]=	{6, -13}
In[20]:=	Kh[Knot[10, 120]][q, t]
Out[20]=	-5 -3 1 3 1 5 3 8 5 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 25 10 23 9 21 9 21 8 19 8 19 7 17 7 q t q t q t q t q t q t q t 8 8 10 8 7 10 6 7 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 6 15 6 15 5 13 5 13 4 11 4 11 3 9 3 q t q t q t q t q t q t q t q t 4 6 4 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t
In[21]:=	ColouredJones[Knot[10, 120], 2][q]
Out[21]=	-34 4 2 15 27 8 70 59 63 158 62 154 226 q - --- + --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 33 32 31 30 29 28 27 26 25 24 23 22 q q q q q q q q q q q q 27 232 239 24 255 192 62 205 109 63 110 38 > --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + 21 20 19 18 17 16 15 14 13 12 11 10 q q q q q q q q q q q q 31 33 7 6 4 -4 > -- - -- + -- + -- - -- + q 9 8 7 6 5 q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10120

10119 10121