10 105

10_104

10_106

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10 105 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X20,8,1,7 X16,5,17,6 X6,15,7,16 X10,17,11,18 X18,9,19,10 X8,14,9,13 X14,20,15,19 X2,12,3,11
Gauss code	1, -10, 2, -1, 4, -5, 3, -8, 7, -6, 10, -2, 8, -9, 5, -4, 6, -7, 9, -3
Dowker-Thistlethwaite code	4 12 16 20 18 2 8 6 10 14
Conway Notation	[21:20:20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 15.1817 A-Polynomial See Data:10 105/A-polynomial

[edit Notes for 10 105's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for 10 105's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-8t^{2}+22t-29+22t^{-1}-8t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 91, 2 }
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-12q^{4}+15q^{3}-15q^{2}+14q-11+7q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+2z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}-3z^{2}+a^{2}+a^{-2}-1$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+11z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}+3az^{7}+6z^{7}a^{-1}+13z^{7}a^{-3}+10z^{7}a^{-5}+a^{2}z^{6}-19z^{6}a^{-2}-3z^{6}a^{-4}+8z^{6}a^{-6}-7z^{6}-8az^{5}-24z^{5}a^{-1}-33z^{5}a^{-3}-13z^{5}a^{-5}+4z^{5}a^{-7}-3a^{2}z^{4}+2z^{4}a^{-2}-9z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}-z^{4}+7az^{3}+18z^{3}a^{-1}+19z^{3}a^{-3}+6z^{3}a^{-5}-2z^{3}a^{-7}+3a^{2}z^{2}+4z^{2}a^{-2}+5z^{2}a^{-4}+3z^{2}a^{-6}+5z^{2}-2az-4za^{-1}-3za^{-3}-za^{-5}-a^{2}-a^{-2}-1$
The A2 invariant	$q^{10}-q^{6}+3q^{4}-2q^{2}+2q^{-2}-3q^{-4}+3q^{-6}-2q^{-8}+2q^{-10}+q^{-12}-2q^{-14}+3q^{-16}-2q^{-18}-q^{-20}+q^{-22}$
The G2 invariant	$q^{46}-2q^{44}+6q^{42}-10q^{40}+13q^{38}-13q^{36}+4q^{34}+18q^{32}-47q^{30}+82q^{28}-97q^{26}+75q^{24}-8q^{22}-95q^{20}+200q^{18}-257q^{16}+230q^{14}-110q^{12}-81q^{10}+262q^{8}-360q^{6}+332q^{4}-170q^{2}-47+233q^{-2}-311q^{-4}+242q^{-6}-67q^{-8}-131q^{-10}+261q^{-12}-259q^{-14}+124q^{-16}+92q^{-18}-293q^{-20}+397q^{-22}-359q^{-24}+181q^{-26}+73q^{-28}-324q^{-30}+472q^{-32}-465q^{-34}+308q^{-36}-48q^{-38}-210q^{-40}+372q^{-42}-381q^{-44}+242q^{-46}-23q^{-48}-174q^{-50}+266q^{-52}-210q^{-54}+44q^{-56}+152q^{-58}-277q^{-60}+283q^{-62}-164q^{-64}-29q^{-66}+203q^{-68}-305q^{-70}+301q^{-72}-199q^{-74}+53q^{-76}+85q^{-78}-173q^{-80}+192q^{-82}-159q^{-84}+96q^{-86}-26q^{-88}-29q^{-90}+57q^{-92}-66q^{-94}+54q^{-96}-33q^{-98}+16q^{-100}+q^{-102}-8q^{-104}+10q^{-106}-10q^{-108}+6q^{-110}-3q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 10_105.

A1 Invariants.

Weight	Invariant
1	$q^{7}-2q^{5}+4q^{3}-4q+3q^{-1}-q^{-3}+3q^{-7}-4q^{-9}+4q^{-11}-3q^{-13}+q^{-15}$
2	$q^{22}-2q^{20}-q^{18}+9q^{16}-7q^{14}-14q^{12}+26q^{10}+q^{8}-38q^{6}+28q^{4}+23q^{2}-44+10q^{-2}+33q^{-4}-26q^{-6}-12q^{-8}+24q^{-10}+5q^{-12}-26q^{-14}+2q^{-16}+36q^{-18}-27q^{-20}-24q^{-22}+46q^{-24}-11q^{-26}-31q^{-28}+28q^{-30}+2q^{-32}-15q^{-34}+8q^{-36}+q^{-38}-3q^{-40}+q^{-42}$
3	$q^{45}-2q^{43}-q^{41}+4q^{39}+6q^{37}-10q^{35}-20q^{33}+15q^{31}+48q^{29}-4q^{27}-88q^{25}-38q^{23}+126q^{21}+108q^{19}-128q^{17}-202q^{15}+86q^{13}+289q^{11}-4q^{9}-330q^{7}-106q^{5}+325q^{3}+209q-278q^{-1}-279q^{-3}+199q^{-5}+314q^{-7}-110q^{-9}-315q^{-11}+27q^{-13}+295q^{-15}+53q^{-17}-249q^{-19}-135q^{-21}+190q^{-23}+212q^{-25}-111q^{-27}-283q^{-29}+9q^{-31}+328q^{-33}+103q^{-35}-326q^{-37}-213q^{-39}+284q^{-41}+279q^{-43}-197q^{-45}-296q^{-47}+100q^{-49}+262q^{-51}-24q^{-53}-192q^{-55}-15q^{-57}+115q^{-59}+27q^{-61}-61q^{-63}-20q^{-65}+30q^{-67}+6q^{-69}-10q^{-71}-3q^{-73}+5q^{-75}+q^{-77}-3q^{-79}+q^{-81}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}-q^{6}+3q^{4}-2q^{2}+2q^{-2}-3q^{-4}+3q^{-6}-2q^{-8}+2q^{-10}+q^{-12}-2q^{-14}+3q^{-16}-2q^{-18}-q^{-20}+q^{-22}$
2,0	$q^{28}-2q^{24}-q^{22}+5q^{20}+5q^{18}-7q^{16}-9q^{14}+9q^{12}+10q^{10}-12q^{8}-14q^{6}+13q^{4}+19q^{2}-14-11q^{-2}+19q^{-4}+5q^{-6}-13q^{-8}-3q^{-10}+9q^{-12}-5q^{-14}-4q^{-16}+11q^{-18}-4q^{-20}-12q^{-22}+12q^{-24}+15q^{-26}-20q^{-28}-10q^{-30}+20q^{-32}+7q^{-34}-17q^{-36}-9q^{-38}+16q^{-40}+4q^{-42}-10q^{-44}-q^{-46}+5q^{-48}+2q^{-50}-3q^{-52}-q^{-54}+q^{-56}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-2q^{18}+2q^{16}+4q^{14}-10q^{12}+8q^{10}+9q^{8}-24q^{6}+18q^{4}+12q^{2}-32+21q^{-2}+13q^{-4}-29q^{-6}+9q^{-8}+14q^{-10}-12q^{-12}-5q^{-14}+6q^{-16}+12q^{-18}-13q^{-20}-9q^{-22}+32q^{-24}-18q^{-26}-18q^{-28}+34q^{-30}-15q^{-32}-17q^{-34}+23q^{-36}-5q^{-38}-10q^{-40}+9q^{-42}-3q^{-46}+q^{-48}$
1,0,0	$q^{13}+q^{9}-q^{7}+3q^{5}-3q^{3}+2q-2q^{-1}+2q^{-3}-2q^{-5}+q^{-7}+q^{-9}-q^{-11}+2q^{-13}-q^{-15}+3q^{-17}-3q^{-19}+3q^{-21}-2q^{-23}-q^{-27}+q^{-29}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-2q^{18}+6q^{16}-10q^{14}+18q^{12}-26q^{10}+33q^{8}-38q^{6}+40q^{4}-36q^{2}+26-13q^{-2}-5q^{-4}+25q^{-6}-45q^{-8}+62q^{-10}-72q^{-12}+77q^{-14}-72q^{-16}+62q^{-18}-45q^{-20}+27q^{-22}-6q^{-24}-12q^{-26}+26q^{-28}-36q^{-30}+39q^{-32}-39q^{-34}+33q^{-36}-25q^{-38}+18q^{-40}-11q^{-42}+6q^{-44}-3q^{-46}+q^{-48}$
1,0	$q^{34}-2q^{30}-2q^{28}+4q^{26}+7q^{24}-2q^{22}-14q^{20}-6q^{18}+19q^{16}+20q^{14}-14q^{12}-33q^{10}-2q^{8}+40q^{6}+22q^{4}-32q^{2}-36+14q^{-2}+42q^{-4}+5q^{-6}-36q^{-8}-17q^{-10}+25q^{-12}+21q^{-14}-17q^{-16}-22q^{-18}+11q^{-20}+25q^{-22}-6q^{-24}-28q^{-26}+31q^{-30}+9q^{-32}-30q^{-34}-20q^{-36}+28q^{-38}+32q^{-40}-17q^{-42}-41q^{-44}+40q^{-48}+18q^{-50}-27q^{-52}-31q^{-54}+8q^{-56}+28q^{-58}+8q^{-60}-15q^{-62}-14q^{-64}+3q^{-66}+10q^{-68}+3q^{-70}-3q^{-72}-3q^{-74}+q^{-78}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-2q^{44}+6q^{42}-10q^{40}+13q^{38}-13q^{36}+4q^{34}+18q^{32}-47q^{30}+82q^{28}-97q^{26}+75q^{24}-8q^{22}-95q^{20}+200q^{18}-257q^{16}+230q^{14}-110q^{12}-81q^{10}+262q^{8}-360q^{6}+332q^{4}-170q^{2}-47+233q^{-2}-311q^{-4}+242q^{-6}-67q^{-8}-131q^{-10}+261q^{-12}-259q^{-14}+124q^{-16}+92q^{-18}-293q^{-20}+397q^{-22}-359q^{-24}+181q^{-26}+73q^{-28}-324q^{-30}+472q^{-32}-465q^{-34}+308q^{-36}-48q^{-38}-210q^{-40}+372q^{-42}-381q^{-44}+242q^{-46}-23q^{-48}-174q^{-50}+266q^{-52}-210q^{-54}+44q^{-56}+152q^{-58}-277q^{-60}+283q^{-62}-164q^{-64}-29q^{-66}+203q^{-68}-305q^{-70}+301q^{-72}-199q^{-74}+53q^{-76}+85q^{-78}-173q^{-80}+192q^{-82}-159q^{-84}+96q^{-86}-26q^{-88}-29q^{-90}+57q^{-92}-66q^{-94}+54q^{-96}-33q^{-98}+16q^{-100}+q^{-102}-8q^{-104}+10q^{-106}-10q^{-108}+6q^{-110}-3q^{-112}+q^{-114}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 105"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{3}-8t^{2}+22t-29+22t^{-1}-8t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}-2z^{4}-z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 91, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{7}-4q^{6}+8q^{5}-12q^{4}+15q^{3}-15q^{2}+14q-11+7q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{6}a^{-2}+2z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}-3z^{2}+a^{2}+a^{-2}-1$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$2z^{9}a^{-1}+2z^{9}a^{-3}+11z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}+3az^{7}+6z^{7}a^{-1}+13z^{7}a^{-3}+10z^{7}a^{-5}+a^{2}z^{6}-19z^{6}a^{-2}-3z^{6}a^{-4}+8z^{6}a^{-6}-7z^{6}-8az^{5}-24z^{5}a^{-1}-33z^{5}a^{-3}-13z^{5}a^{-5}+4z^{5}a^{-7}-3a^{2}z^{4}+2z^{4}a^{-2}-9z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}-z^{4}+7az^{3}+18z^{3}a^{-1}+19z^{3}a^{-3}+6z^{3}a^{-5}-2z^{3}a^{-7}+3a^{2}z^{2}+4z^{2}a^{-2}+5z^{2}a^{-4}+3z^{2}a^{-6}+5z^{2}-2az-4za^{-1}-3za^{-3}-za^{-5}-a^{2}-a^{-2}-1$

Vassiliev invariants

V2 and V3:

(-1, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-4$ $0$ $8$ ${\frac {82}{3}}$ ${\frac {62}{3}}$ $0$ $32$ $32$ $0$ $-{\frac {32}{3}}$ $0$ $-{\frac {328}{3}}$ $-{\frac {248}{3}}$ $-{\frac {1951}{30}}$ ${\frac {1022}{15}}$ $-{\frac {7022}{45}}$ ${\frac {703}{18}}$ $-{\frac {511}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 10 105. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                     1 1 13                   3   -3 11                 5 1   4 9               7 3     -4 7             8 5       3 5           7 7         0 3         7 8           -1 1       5 8             3 -1     2 6               -4 -3   1 5                 4 -5   2                   -2 -7 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 105]]
Out[2]=	10
In[3]:=	PD[Knot[10, 105]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[20, 8, 1, 7], X[16, 5, 17, 6], X[6, 15, 7, 16], X[10, 17, 11, 18], X[18, 9, 19, 10], X[8, 14, 9, 13], X[14, 20, 15, 19], X[2, 12, 3, 11]]
In[4]:=	GaussCode[Knot[10, 105]]
Out[4]=	GaussCode[1, -10, 2, -1, 4, -5, 3, -8, 7, -6, 10, -2, 8, -9, 5, -4, 6, -7, 9, -3]
In[5]:=	BR[Knot[10, 105]]
Out[5]=	BR[5, {1, 1, -2, 1, 3, 2, 2, -4, -3, 2, -3, -4}]
In[6]:=	alex = Alexander[Knot[10, 105]][t]
Out[6]=	-3 8 22 2 3 -29 + t - -- + -- + 22 t - 8 t + t 2 t t
In[7]:=	Conway[Knot[10, 105]][z]
Out[7]=	2 4 6 1 - z - 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 105], Knot[11, NonAlternating, 163]}
In[9]:=	{KnotDet[Knot[10, 105]], KnotSignature[Knot[10, 105]]}
Out[9]=	{91, 2}
In[10]:=	J=Jones[Knot[10, 105]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 -11 + q - -- + - + 14 q - 15 q + 15 q - 12 q + 8 q - 4 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 105]}
In[12]:=	A2Invariant[Knot[10, 105]][q]
Out[12]=	-10 -6 3 2 2 4 6 8 10 12 q - q + -- - -- + 2 q - 3 q + 3 q - 2 q + 2 q + q - 4 2 q q 14 16 18 20 22 2 q + 3 q - 2 q - q + q
In[13]:=	Kauffman[Knot[10, 105]][a, z]
Out[13]=	2 2 2 -2 2 z 3 z 4 z 2 3 z 5 z 4 z -1 - a - a - -- - --- - --- - 2 a z + 5 z + ---- + ---- + ---- + 5 3 a 6 4 2 a a a a a 3 3 3 3 4 4 2 2 2 z 6 z 19 z 18 z 3 4 z 8 z 3 a z - ---- + ---- + ----- + ----- + 7 a z - z + -- - ---- - 7 5 3 a 8 6 a a a a a 4 4 5 5 5 5 9 z 2 z 2 4 4 z 13 z 33 z 24 z 5 ---- + ---- - 3 a z + ---- - ----- - ----- - ----- - 8 a z - 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 6 8 z 3 z 19 z 2 6 10 z 13 z 6 z 7 7 z + ---- - ---- - ----- + a z + ----- + ----- + ---- + 3 a z + 6 4 2 5 3 a a a a a a 8 8 9 9 8 7 z 11 z 2 z 2 z 4 z + ---- + ----- + ---- + ---- 4 2 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 105]], Vassiliev[3][Knot[10, 105]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 105]][q, t]
Out[15]=	3 1 2 1 5 2 6 5 q 8 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 8 q t + 7 q t + 7 q t + 8 q t + 5 q t + 7 q t + 3 q t + 11 4 11 5 13 5 15 6 5 q t + q t + 3 q t + q t