The Non Alternating Knot 10₁₃₂

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X₈₄₉₃ X_5,12,6,13 X_15,18,16,19 X_9,16,10,17 X_17,10,18,11 X_13,20,14,1 X_19,14,20,15 X_11,6,12,7 X₂₈₃₇

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

DT (Dowker-Thistlethwaite) Code: 4 8 -12 2 -16 -6 -20 -18 -10 -14

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 1 2 3 / NotAvailable 1

Alexander Polynomial: t^-2 - t^-1 + 1 - t + t²

Conway Polynomial: 1 + 3z² + z⁴

Other knots with the same Alexander/Conway Polynomial: {5₁, ...}

Determinant and Signature: {5, 0}

Jones Polynomial: - q^-7 + q^-6 - q^-5 + q^-4 + q^-2

Other knots (up to mirrors) with the same Jones Polynomial: {5₁, ...}

A2 (sl(3)) Invariant: - q^-22 - q^-20 - q^-18 + q^-14 + q^-12 + 2q^-10 + q^-8 + q^-6

HOMFLY-PT Polynomial: 3a⁴ + 4a⁴z² + a⁴z⁴ - 2a⁶ - a⁶z²

Kauffman Polynomial: - az - a²z² - 4a³z + 9a³z³ - 6a³z⁵ + a³z⁷ + 3a⁴ - 7a⁴z² + 10a⁴z⁴ - 6a⁴z⁶ + a⁴z⁸ - 8a⁵z + 19a⁵z³ - 12a⁵z⁵ + 2a⁵z⁷ + 2a⁶ - 6a⁶z² + 10a⁶z⁴ - 6a⁶z⁶ + a⁶z⁸ - 5a⁷z + 10a⁷z³ - 6a⁷z⁵ + a⁷z⁷

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {3, -5}

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=0 is the signature of 10₁₃₂. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -1 1 1

j = -3 1

j = -5 1 2

j = -7 1

j = -9 1 1

j = -11 1 1

j = -13

j = -15 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-21 - q^-20 - q^-19 + 2q^-18 - q^-17 - q^-16 + 2q^-15 - 2q^-14 + 2q^-12 - 3q^-11 + q^-10 + 2q^-9 - 3q^-8 + 2q^-7 + 2q^-6 - 4q^-5 + 3q^-4 + q^-3 - 3q^-2 + 2q^-1 + 1 - q

3 - q^-42 + q^-41 + q^-40 - 2q^-38 + 2q^-36 - 2q^-34 + 3q^-32 - q^-31 - 3q^-30 + q^-29 + 4q^-28 - q^-27 - 5q^-26 + q^-25 + 5q^-24 - q^-23 - 5q^-22 + 5q^-20 - 4q^-18 - q^-17 + 4q^-16 + q^-15 - 3q^-14 - 2q^-13 + 3q^-12 + q^-11 - 2q^-10 - 2q^-9 + 3q^-8 + 2q^-7 - q^-6 - 2q^-5 + q^-4 + 2q^-3 - q^-1

4 q^-70 - q^-69 - q^-68 + 3q^-65 - q^-64 - q^-63 - q^-62 - q^-61 + 4q^-60 - q^-59 - q^-58 - 2q^-57 + 5q^-55 - q^-54 - 2q^-53 - 4q^-52 + q^-51 + 6q^-50 + q^-49 - 2q^-48 - 6q^-47 + 6q^-45 + 2q^-44 - 2q^-43 - 5q^-42 + 5q^-40 + q^-39 - 3q^-38 - 3q^-37 + q^-36 + 6q^-35 - q^-34 - 5q^-33 - 2q^-32 + 2q^-31 + 8q^-30 - 2q^-29 - 7q^-28 - 2q^-27 + 2q^-26 + 10q^-25 - 2q^-24 - 9q^-23 - 2q^-22 + 2q^-21 + 11q^-20 - 3q^-19 - 10q^-18 - q^-17 + 3q^-16 + 11q^-15 - 3q^-14 - 10q^-13 + q^-12 + 3q^-11 + 10q^-10 - 3q^-9 - 9q^-8 + 2q^-6 + 9q^-5 - 2q^-4 - 6q^-3 - q^-2 + 5 - q² - q³ - q⁴ + q⁵

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, 132]]

Out[2]=
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[15, 18, 16, 19], > X[9, 16, 10, 17], X[17, 10, 18, 11], X[13, 20, 14, 1], X[19, 14, 20, 15], > X[11, 6, 12, 7], X[2, 8, 3, 7]]

In[3]:=
GaussCode[Knot[10, 132]]

Out[3]=
GaussCode[1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, > 7]

In[4]:=
DTCode[Knot[10, 132]]

Out[4]=
DTCode[4, 8, -12, 2, -16, -6, -20, -18, -10, -14]

In[5]:=
br = BR[Knot[10, 132]]

Out[5]=
BR[4, {1, 1, 1, -2, -1, -1, -2, -3, 2, -3, -3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 132]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 132]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 132]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 1, 2, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 132]][t]

Out[10]=
-2 1 2 1 + t - - - t + t t

In[11]:=
Conway[Knot[10, 132]][z]

Out[11]=
2 4 1 + 3 z + z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[5, 1], Knot[10, 132]}

In[13]:=
{KnotDet[Knot[10, 132]], KnotSignature[Knot[10, 132]]}

Out[13]=
{5, 0}

In[14]:=
Jones[Knot[10, 132]][q]

Out[14]=
-7 -6 -5 -4 -2 -q + q - q + q + q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[5, 1], Knot[10, 132]}

In[16]:=
A2Invariant[Knot[10, 132]][q]

Out[16]=
-22 -20 -18 -14 -12 2 -8 -6 -q - q - q + q + q + --- + q + q 10 q

In[17]:=
HOMFLYPT[Knot[10, 132]][a, z]

Out[17]=
4 6 4 2 6 2 4 4 3 a - 2 a + 4 a z - a z + a z

In[18]:=
Kauffman[Knot[10, 132]][a, z]

Out[18]=
4 6 3 5 7 2 2 4 2 6 2 3 a + 2 a - a z - 4 a z - 8 a z - 5 a z - a z - 7 a z - 6 a z + 3 3 5 3 7 3 4 4 6 4 3 5 5 5 > 9 a z + 19 a z + 10 a z + 10 a z + 10 a z - 6 a z - 12 a z - 7 5 4 6 6 6 3 7 5 7 7 7 4 8 6 8 > 6 a z - 6 a z - 6 a z + a z + 2 a z + a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 132]], Vassiliev[3][Knot[10, 132]]}

Out[19]=
{3, -5}

In[20]:=
Kh[Knot[10, 132]][q, t]

Out[20]=
-3 1 1 1 1 1 1 1 1 2 1 q + - + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + --- q 15 7 11 6 11 5 9 4 7 4 9 3 5 3 5 2 q t q t q t q t q t q t q t q t q t

In[21]:=
ColouredJones[Knot[10, 132], 2][q]

Out[21]=
-21 -20 -19 2 -17 -16 2 2 2 3 -10 1 + q - q - q + --- - q - q + --- - --- + --- - --- + q + 18 15 14 12 11 q q q q q 2 3 2 2 4 3 -3 3 2 > -- - -- + -- + -- - -- + -- + q - -- + - - q 9 8 7 6 5 4 2 q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₃₂

10₁₃₁
10₁₃₃

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-21 - q^-20 - q^-19 + 2q^-18 - q^-17 - q^-16 + 2q^-15 - 2q^-14 + 2q^-12 - 3q^-11 + q^-10 + 2q^-9 - 3q^-8 + 2q^-7 + 2q^-6 - 4q^-5 + 3q^-4 + q^-3 - 3q^-2 + 2q^-1 + 1 - q
3	- q^-42 + q^-41 + q^-40 - 2q^-38 + 2q^-36 - 2q^-34 + 3q^-32 - q^-31 - 3q^-30 + q^-29 + 4q^-28 - q^-27 - 5q^-26 + q^-25 + 5q^-24 - q^-23 - 5q^-22 + 5q^-20 - 4q^-18 - q^-17 + 4q^-16 + q^-15 - 3q^-14 - 2q^-13 + 3q^-12 + q^-11 - 2q^-10 - 2q^-9 + 3q^-8 + 2q^-7 - q^-6 - 2q^-5 + q^-4 + 2q^-3 - q^-1
4	q^-70 - q^-69 - q^-68 + 3q^-65 - q^-64 - q^-63 - q^-62 - q^-61 + 4q^-60 - q^-59 - q^-58 - 2q^-57 + 5q^-55 - q^-54 - 2q^-53 - 4q^-52 + q^-51 + 6q^-50 + q^-49 - 2q^-48 - 6q^-47 + 6q^-45 + 2q^-44 - 2q^-43 - 5q^-42 + 5q^-40 + q^-39 - 3q^-38 - 3q^-37 + q^-36 + 6q^-35 - q^-34 - 5q^-33 - 2q^-32 + 2q^-31 + 8q^-30 - 2q^-29 - 7q^-28 - 2q^-27 + 2q^-26 + 10q^-25 - 2q^-24 - 9q^-23 - 2q^-22 + 2q^-21 + 11q^-20 - 3q^-19 - 10q^-18 - q^-17 + 3q^-16 + 11q^-15 - 3q^-14 - 10q^-13 + q^-12 + 3q^-11 + 10q^-10 - 3q^-9 - 9q^-8 + 2q^-6 + 9q^-5 - 2q^-4 - 6q^-3 - q^-2 + 5 - q² - q³ - q⁴ + q⁵

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 132]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[15, 18, 16, 19], > X[9, 16, 10, 17], X[17, 10, 18, 11], X[13, 20, 14, 1], X[19, 14, 20, 15], > X[11, 6, 12, 7], X[2, 8, 3, 7]]
In[3]:=	GaussCode[Knot[10, 132]]
Out[3]=	GaussCode[1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, > 7]
In[4]:=	DTCode[Knot[10, 132]]
Out[4]=	DTCode[4, 8, -12, 2, -16, -6, -20, -18, -10, -14]
In[5]:=	br = BR[Knot[10, 132]]
Out[5]=	BR[4, {1, 1, 1, -2, -1, -1, -2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 132]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 132]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 132]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 132]][t]
Out[10]=	-2 1 2 1 + t - - - t + t t
In[11]:=	Conway[Knot[10, 132]][z]
Out[11]=	2 4 1 + 3 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[5, 1], Knot[10, 132]}
In[13]:=	{KnotDet[Knot[10, 132]], KnotSignature[Knot[10, 132]]}
Out[13]=	{5, 0}
In[14]:=	Jones[Knot[10, 132]][q]
Out[14]=	-7 -6 -5 -4 -2 -q + q - q + q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[5, 1], Knot[10, 132]}
In[16]:=	A2Invariant[Knot[10, 132]][q]
Out[16]=	-22 -20 -18 -14 -12 2 -8 -6 -q - q - q + q + q + --- + q + q 10 q
In[17]:=	HOMFLYPT[Knot[10, 132]][a, z]
Out[17]=	4 6 4 2 6 2 4 4 3 a - 2 a + 4 a z - a z + a z
In[18]:=	Kauffman[Knot[10, 132]][a, z]
Out[18]=	4 6 3 5 7 2 2 4 2 6 2 3 a + 2 a - a z - 4 a z - 8 a z - 5 a z - a z - 7 a z - 6 a z + 3 3 5 3 7 3 4 4 6 4 3 5 5 5 > 9 a z + 19 a z + 10 a z + 10 a z + 10 a z - 6 a z - 12 a z - 7 5 4 6 6 6 3 7 5 7 7 7 4 8 6 8 > 6 a z - 6 a z - 6 a z + a z + 2 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 132]], Vassiliev[3][Knot[10, 132]]}
Out[19]=	{3, -5}
In[20]:=	Kh[Knot[10, 132]][q, t]
Out[20]=	-3 1 1 1 1 1 1 1 1 2 1 q + - + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + --- q 15 7 11 6 11 5 9 4 7 4 9 3 5 3 5 2 q t q t q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 132], 2][q]
Out[21]=	-21 -20 -19 2 -17 -16 2 2 2 3 -10 1 + q - q - q + --- - q - q + --- - --- + --- - --- + q + 18 15 14 12 11 q q q q q 2 3 2 2 4 3 -3 3 2 > -- - -- + -- + -- - -- + -- + q - -- + - - q 9 8 7 6 5 4 2 q q q q q q q q

The Non Alternating Knot 10132

The Non Alternating Knot 10₁₃₂